UNIVERSITY  OF  CALIFORNIA 
AT  LOS  ANGELES 


ROLF  HOFFMANN 


A  HISTORY 


OF 


vPANESE  MATHEMATICS 


BY 


DAVID  EUGENE  SMITH 


AND 


YOSHIO  MIKAMI 


CHICAGO 

THE  OPEN  COURT  PUBLISHING  COMPANY 
1914 


Printed  by  W.   Drugulin,  Leipzig 


Mathematics/ 
Sckrnoes 


PREFACE 

Although   for   nearly    a    century  the  greatest   mathematical 

classics    of  India   have   been  known  to  western  scholars,    and 

several    of  the   more  important  works  of  the  Arabs  for  even 

longer,  the  mathematics  of  China  and  Japan  has  been  closed 

to    all   European   and  American   students   until   very   recently. 

Even    now    we    have    not    a    single    translation    of  a    Chinese 

treatise  upon  the  subject,  and  it  is  only  within  the  last  dozen 

years    that    the    contributions    of  the   native  Japanese   school 

have    become    known    in    the  West  even    by    name.      At   the 

&  second  International  Congress  of  Mathematicians,  held  at  Paris 

^  in  1900,  Professor  Fujisawa  of  the  Imperial  University  of  Tokio 

gave  a  brief  address   upon  Mathematics    of  the   old  Japanese 

p  School,  and  this  may  be  taken  as  the  first  contribution  to  the 

j  history   of  mathematics   made  by   a  native  of  that  country  in 

J  a  European    language.     The   next   effort  of  this  kind  showed 

?   itself  in  occasional  articles  by  Baron  Kikuchi,  as  in  the  Nieuw 

1    Archief  voor   Wiskunde,    some  of  which  were  based  upon  his 

contributions    in  Japanese   to   one  of  the  scientific  journals  of 

Tokio.     But  the  only  serious  attempt  made  up  to  the  present 

time    to    present    a   well-ordered  history   of  the   subject   in   a 

European   language    is   to   be  found  in  the  very  commendable 

papers  by  T.  Hayashi,   of  the  Imperial  University  at  Sendai. 

The    most   important  of  these   have   appeared  in   the  Nieuw 

Archief  voor   Wiskunde,   and   to  them  the  authors  are  much 

indebted. 

Having  made  an  extensive  collection  of  mathematical  manu- 
scripts, early  printed  works,  and  early  instruments,  and  having 


154988 


IV  Preface. 

brought  together  most  of  the  European  literature  upon  the 
subject  and  embodied  it  in  a  series  of  lectures  for  my  classes 
in  the  history  of  mathematics,  I  welcomed  the  suggestion  of 
Dr.  Carus  that  I  join  with  Mr.  Mikami  in  the  preparation  of 
the  present  work.  Mr.  Mikami  has  already  made  for  himself 
an  enviable  reputation  as  an  authority  upon  the  wasan  or 
native  Japanese  mathematics,  and  his  contributions  to  the 
Bibliotheca  Mathematica  have  attracted  the  attention  of  western 
scholars.  He  has  also  published,  as  a  volume  of  the  Abhand- 
lungen  zur  Geschichte  der  Mathcmatik,  a  work  entitled  Mathe- 
matical Papers  from  the  Far  East.  Moreover  his  labors  with 
the  learned  T.  Endo,  the  greatest  of  the  historians  of  Japanese 
mathematics,  and  his  consequent  familiarity  with  the  classics 
of  his  country,  eminently  fit  him  for  a  work  of  this  nature. 

Our  labors  have  been  divided  in  the  manner  that  the  cir- 
cumstances would  suggest.  For  the  European  literature,  the 
general  planning  of  the  work,  and  the  final  writing  of  the  text, 
the  responsibility  has  naturally  fallen  to  a  considerable  extent 
upon  me.  For  the  furnishing  of  the  Japanese  material,  the 
initial  translations,  the  scholarly  search  through  the  excellent 
library  of  the  Academy  of  Sciences  of  Tokio,  where  Mr.  Endo 
is  librarian,  and  the  further  examination  of  the  large  amount  of 
native  secondary  material,  the  responsibility  has  been  Mr.  Mi- 
kami's.  To  his  scholarship  and  indefatigable  labors  I  am  in- 
debted for  more  material  than  could  be  used  in  this  work, 
and  whatever  praise  our  efforts  may  merit  should  be  awarded 
in  large  measure  to  him. 

The  aim  in  writing  this  work  has  been  to  give  a  brief 
survey  of  the  leading  features  in  the  development  of  the  ivasan. 
It  has  not  seemed  best  to  enter  very  fully  into  the  details  of 
demonstration  or  into  the  methods  of  solution  employed  by 
the  great  writers  whose  works  are  described.  This  would  not 
be  done  in  a  general  history  of  European  mathematics,  and 
there  is  no  reason  why  it  should  be  done  here,  save  in  cases 
where  some  peculiar  feature  is  under  discussion.  Undoubtedly 
several  names  of  importance  have  been  omitted,  and  at  least 
a  score  of  names  that  might  properly  have  had  mention  have 


Preface.  V 

been  the  subject  of  correspondence  between  the  authors  for 
the  past  year.  But  on  the  whole  it  may  be  said  that  most 
of  those  writers  in  whose  works  European  scholars  are  likely 
to  have  much  interest  have  been  mentioned. 

It  is  the  hope  of  the  authors  that  this  brief  history  may 
serve  to  show  to  the  West  the  nature  of  the  mathematics 
that  wras  indigenous  to  Japan,  and  to  strengthen  the  bonds 
that  unite  the  scholars  of  the  world  through  an  increase  in 
knowledge  of  and  respect  for  the  scientific  attainments  of  a 
people  whose  progress  in  the  past  four  centuries  has  been 
one  of  the  marvels  of  history. 

It  is  only  just  to  mention  at  this  time  the  generous  assistance 
rendered  by  Mr.  Leslie  Leland  Locke,  one  of  my  graduate 
students  in  the  history  of  mathematics,  who  made  in  my 
library  the  photographs  for  all  of  the  illustrations  used  in  this 
work.  His  intelligent  and  painstaking  efforts  to  carry  out  the 
wishes  of  the  authors  have  resulted  in  a  series  of  illustrations 
that  not  merely  elucidate  the  text,  but  give  a  visual  idea  of 
the  genius  of  the  Japanese  mathematics  that  words  alone 
cannot  give.  To  him  I  take  pleasure  in  ascribing  the  credit 
for  this  arduous  labor,  and  in  expressing  the  thanks  of  the 
authors. 

Teachers  College, 

Columbia  University,  David  Eugene  Smith. 

New  York  City, 

December  r,  1913. 


VOCABULARY  FOR  REFERENCE 

The  following  brief  vocabulary  will  be  convenient  for  reference  in  con- 
sidering some  of  the  Japanese  titles: 

ho,  method  or  theory.     Synonym  of  jutsu.     It  is  found  in  expressions  like 

shosa  ho  (method  of  differences). 
hyo,  table. 
jutsu,    method    or    theory.     Synonym    of   ho,     It    is    found    in   words    like 

kaku  jutsu    (polygonal    theory)    and    tatsujutsu    (method   of  expanding  a 

root  of  a  literal  equation). 
kit  a  treatise. 

roku,  a  treatise.     Synonym  of  ki. 
sampo,  mathematical  treatise,  or  mathematical  lules. 

sangi,  rods  used  in  computing,   and  as  numerical  coefficients  in  equations 
soroban,  the  Japanese  abacus. 

tengen,  celestial  element.     The  Japanese  name  for  the  Chinese  algebra. 
tenzan,  the  algebra  of  the  Seki  school. 
wasan,   the    native  Japanese  mathematics  as  distinguished  from  the  yosan, 

the  European  mathematics. 
yenri,  circle  principle.     A  term  applied  to  the  native  calculus  of  Japan. 

In  Japanese  proper  names  the  surname  is  placed  first  in  accordance  with 
the  native  custom,  excepting  in  the  cases  of  persons  now  living  and  who 
follow  the  European  custom  of  placing  the  surname  last. 


CONTENTS 

CHAPTER  PAGE 

I.    The  Earliest  Period I 

II.    The  Second  Period 7 

III.  The  Development  of  the  Soroban 18 

IV.  The  Sangi  applied  to  Algebra  .    .    .     • 47 

V.   The  Third  Period 59 

VI.    Seki  Kowa 91 

VII.    Seki's  Contemporaries  and  possible  Western  Influences     ....  128 

VIII.    The  Yenri  or  Circle  Principle H3 

IX.   The  Eighteenth  Century 163 

X.   Ajima  Chokuyen 195 

XL   The  Opening  of  the  Nineteenth  Century 206 

XII.    Wada  Nei 220 

XIII.  The  Close  of  the  Old  Wasan 230 

XIV.  The  Introduction  of  Occidental  Mathematics 252 

Index  .                                                                   281 


CHAPTER  I. 

»  '  The  Earliest  Period. 

The  history  of  Japanese  mathematics,  from  the  most  remote 
times  to  the  present,  may  be  divided  into  six  fairly  distinct 
periods.  Of  these  the  first  extended  from  the  earliest  ages  to 
552  T,  a  period  that  was  influenced  only  indirectly  if  at  all  by 
Chinese  mathematics.  The  second  period  of  approximately  a 
thousand  years  (552 — 1600)  was  characterized  by  the  influx 
of  Chinese  learning,  first  through  Korea  and  then  direct  from 
China  itself,  by  some  resulting  native  development,  and  by  a 
season  of  stagnation  comparable  to  the  Dark  Ages  of  Europe. 
The  third  period  was  less  than  a  century  in  duration,  extend- 
ing from  about  1600  to  the  beginning  of  Seki's  influence  (about 
1675).  This  may  be  called  the  Renaissance  period  of  Japanese 
mathematics,  since  it  saw  a  new  and  vigorous  importation  of 
Chinese  science,  the  revival  of  native  interest  through  the  efforts 
of  the  immediate  predecessors  of  Seki,  and  some  slight  intro- 
duction of  European  learning  through  the  early  Dutch  traders 
and  through  the  Jesuits.  The  fourth  period,  also  about  a  century 
in  length  (1675  to  1775)  may  be  compared  to  the  synchro- 
nous period  in  Europe.  Just  as  the  initiative  of  Descartes, 
Newton,  and  Leibnitz  prepared  the  way  for  the  labors  of  the 
Bernoullis,  Euler,  Laplace,  D'Alembert,  and  their  contemporaries 
of  the  eighteenth  century,  so  the  work  of  the  great  Japanese 
teacher,  Seki,  and  of  his  pupil  Takebe,  made  possible  a  note- 
worthy development  of  the  wasan 2  of  Japan  during  the  same 

1  All  dates  are  expressed  according  to  the  Christian  calendar  and   are  to 
be  taken  as  after  Christ  unless  the  contrary  is  stated. 

2  The  native  mathematics,  from  Wa  (Japan)   and  san  (mathematics).     The 
word  is  modern,  having  been  employed  to  distinguish  the  native  theory  from 
the  western  mathematics,  the  yosan. 

I 


2  I.  The  Earliest  Period. 

century.  The  fifth  period,  which  might  indeed  be  joined  with 
the  fourth,  but  which  differs  from  it  much  as  the  nineteenth 
century  of  European  mathematics  differs  from  the  eighteenth, 
extended  from  1775  to  1868,  the  date  of  the  opening  of  Japan 
to  the  Western  World.  This  is  the  period  of  the  culmination 
of  native  Japanese  mathematics,  as  influenced  more  or  less  by 
the  European  learning  that  managed  to  find  some  entrance 
through  the  Dutch  trading  station  at  Nagasaki  and  through 
the  first  Christian  missionaries.  The  sixth  and  final  period 
begins  with  the  opening  of  Japan  to  intercourse  with  other 
countries  and  extends  to  the  present  time,  a  period  of  marvelous 
change  in  government,  in  ideals,  in  art,  in  industry,  in  edu- 
cation, in  mathematics  and  the  sciences  generally,  and  in  all 
that  makes  a  nation  great.  With  these  stupendous  changes 
of  the  present,  that  have  led  Japan  to  assume  her  place  among 
the  powers  of  the  world,  there  has  necessarily  come  both  loss 
and  gain.  Just  as  the  world  regrets  the  apparent  submerging 
of  the  exquisite  native  art  of  Japan  in  the  rising  tide  of  com- 
mercialism, so  the  student  of  the  history  of  mathematics  must 
view  with  sorrow  the  necessary  decay  of  the  wasan  and  the 
reduction  or  the  elevation  of  this  noble  science  to  the  general 
cosmopolitan  level.  The  mathematics  of  the  present  in  Japan 
is  a  broader  science  than  that  of  the  past;  but  it  is  no  longer 
Japanese  mathematics, — it  is  the  mathematics  of  the  world. 

It  is  now  proposed  to  speak  of  the  first  period,  extending 
from  the  most  remote  times  to  552.  From  the  nature  of  the 
case,  however,  little  exact  information  can  be  expected  of  this 
period.  It  [is  like  seeking  for  the  early  history  of  England 
from  native  sources,  excluding  all  information  transmitted 
through  Roman  writers.  Egypt  developed  a  literature  in 
very  remote  times,  and  recorded  it  upon  her  monuments 
and  upon  papyrus  rolls,  and  Babylon  wrote  her  records  upon 
both  stone  and  clay;  but  Japan  had  no  early  literature,  and  if 
she  possessed  any  ancient  written  records  they  have  long  since 
perished. 

It  was  not  until  the  fifteenth  year  of  the  Emperor  Ojin  (284), 
so  the  story  goes,  that  Chinese  ideograms,  making  their  way 


I.  The  Earliest  Period.  3 

through  Korea,  were  first  introduced  into  Japan.  Japanese 
nobles  now  began  to  learn  to  read  and  write,  a  task  of  enor- 
mous difficulty  in  the  Chinese  system.  But  the  records  them- 
selves have  long  since  perished,  and  if  they  contained  any 
knowledge  of  mathematics,  or  if  any  mathematics  from  China 
at  that  time  reached  the  shores  of  Japan,  all  knowledge  of 
this  fact  has  probably  gone  forever.  Nevertheless  there  is 
always  preserved  in  the  language  of  a  people  a  great  amount 
of  historical  material,  and  from  this  and  from  folklore  and  tra- 
dition we  can  usually  derive  some  little  knowledge  of  the  early 
life  and  customs  and  number-science  of  any  nation. 

So  it  is  with  Japan.  There  seems  to  have  been  a  number 
mysticism  there  as  in  all  other  countries.  There  was  the 
usual  reaching  out  after  the  unknown  in  the  study  of  the  stars, 
of  the  elements,  and  of  the  essence  of  life  and  the  meaning 
of  death.  The  general  expression  of  wonder  that  comes  from 
the  study  of  number,  of  forms,  and  of  the  arrangements  of 
words  and  objects,  is  indicated  in  the  language  and  the  tradi- 
tions of  Japan  as  in  the  language  and  traditions  of  all  other 
peoples.  Thus  we  know  that  the  Jindai  monji,  "letters  of  the 
era  of  the  gods",1  go  back  to  remote  times,  and  this  suggests 
an  early  cabala,  very  likely  with  its  usual  accompaniment  of 
number  values  to  the  letters;  but  of  positive  evidence  of  this 
fact  we  have  none,  and  we  are  forced  to  rely  at  present  only 
upon  conjecture.2 

Practically  only  one   definite   piece  of  information  has  come 

1  Nothing   definite   is   known   as   to   these   letters.     They   may  have   been 
different   alphabetic   forms.     Monji  (or  moji)  means   letters,  Jin   is    god,   and 
dai  is  the  age  or  era.     The  expression  may  also  be  rendered  "letters  of  the 
age   of   heros",  using   the   term    hero   to    mean   a  mythological  semi-divinity, 
as  it  is  used  in  early  Greek  lore. 

2  There  is  here,  howewer,  an    excellent   field   for   some   Japanese   scholar 
to  search   the   native   folklore   for  new   material.     Our  present  knowledge  of 
the  Jindai  comes  chiefly  from    a   chapter  in  the  Nihon-gi  (Records  of  Japan) 
entitled  Jindai  no  Maki  (Records  of  the  Gods'  Age),  written  by  Prince  Toneri 
Shinno  in  720.     This  is  probably  based  upon  early  legends  handed  down  by 
the  Kataribe,   a  class  of  men  who   in   ancient   times   transmitted  the  legends 
orally,  somewhat  like  the  old  English  bards. 

i* 


4  I.  The  Earliest  Period. 

down  to  us  concerning  the  very  early  mathematics  of  Japan, 
and  this  relates  to  the  number  system.  Tradition  tells  us  that 
in  the  reign  of  Izanagi-no-Mikoto,  the  ancestor  of  the  Mikados, 
long  before  the  unbroken  dynasty  was  founded  by  Jimmu 
(660  B.  C),  a  system  of  numeration  was  known  that  extended 
to  very  high  powers  of  ten,  and  that  embodied  essentially  the 
exponential  law  used  by  Archimedes  in  his  Sand  Reckoner*1  that 

a'"an  =  am+*. 

In  this  system  the  number  names  were  not  those  of  the  present, 
but  the  system  may  have  been  the  same,  although  modern 
Japanese  anthropologists  have  serious  doubts  upon  this  matter. 
The  following  table2  has  been  given  as  representing  the  ancient 
system,  and  it  is  inserted  as  a  possibility,  but  the  whole  matter 
is  in  need  of  further  investigation: 


Ancient 

Modern 

Ancient 

Modern 

I  hito 

ichi 

TOO 

momo 

hyaku 

2  futa 

ni 

IOOO 

chi 

sen 

3  mi 

san 

IOOOO 

yorozu 

man 

4  yo 

shi 

IOOOOO 

so  yorozu 

jiu  man 

5  itsu 

g° 

lOOOOOO 

momo  yorozu 

hyaku  man 

6  mu 

roku 

lOOOOOOO 

chi  yorozu 

sen  man 

7  nana 

shichi 

IOOOOO  OOO 

yorozu  yorozu 

oku 

8  ya 

hachi 

IOOOOOOOOO 

so  yorozu  yorozu 

jiu  oku 

9  koko 

ku 

10  to 

jiu 

^,  De  harena  numero,  as  it  appears  in  Basel  edition  of  1544. 
2  ENDO,  T.,  Dai  Nikon  Sitgaku  Shi  (History  of  Japanese  mathematics,  in 
Japanese.  Tokio  1896,  Book  I,  pp.  3  —  5;  hereafter  referred  to  as  ENDU). 
See  also  KNOTT,  C.  G.,  The  Abacus  in  its  historic  and  scientific  aspects,  in  the 
Transactions  of  the  Asiatic  Society  of  Japan,  Yokohama  1 886,  vol.  XIV,  p.  38; 
hereafter  referred  to  as  Knott.  Another  interesting  form  of  counting  is  still 
in  use  in  Japan,  and  is  more  closely  connected  with  the  ancient  one  than 
is  the  common  form  above  given.  It  is  as  follows:  (I)  hitotsu,  (2)  futatsu, 
(3)  mittsu,  (4)  yottsu,  (5)  itsutsu,  (6)  muttsu,  (7)  nanatsu,  (8)  yattsu,  (9^  koko- 
notsu,  (10)  to.  Still  another  form  at  present  in  use,  and  also  related  to  the 
ancient  one,  is  as  follows:  (l)  hi,  (2)  fu,  (3)  mi,  (4)  yo,  (5)  itsu,  (6)  mvi, 
(7)  nana,  (8)  ya,  (9)  kono,  (10)  to.  Each  of  these  forms  is  used  only  in 
counting,  not  in  naming  numbers,  and  their  persistence  may  be  compared 


I.  The  Earliest  Period.  5 

The  interesting  features  of  the  ancient  system  are  the  deci- 
mal system  and  the  use  of  the  word  yorozu,  which  now  means 
10000.  This,  however,  may  be  a  meaning  that  came  with  the 
influx  of  Chinese  learning,  and  we  are  not  at  all  certain  that 
in  ancient  Japanese  it  stood  for  the  Greek  myriad. x  The  use 
of  yorozu  for  10000  was  adopted  in  later  times  when  the 
number  names  came  to  be  based  upon  Chinese  roots,  and  it 
may  possibly  have  preceded  the  entry  of  Chinese  learning  in 
historic  times.  Thus  IQS  was  not  "hundred  thousand"2  in  this 
later  period,  but  "ten  myriads", 3  and  our  million*  is  a  hundred 
myriads,  s  Now  this  system  of  numeration  by  myriads  is  one 
of  the  frequently  observed  evidences  of  early  intercourse  between 
the  scholars  of  the  East  and  the  West.  Trades  people  and 
the  populace  at  large  did  not  need  such  large  numbers,  but 
to  the  scholar  they  were  significant.  When,  therefore,  we  find 
the  myriad  as  the  base  of  the  Greek  system,6  and  find  it 
more  or  less  in  use  in  India,7  and  know  that  it  still  persists 
in  China,8  and  see  it  systematically  used  in  the  ancient  Japa- 
nese system  as  well  as  in  the  modern  number  names,  we  are 

with  that  of  the  "counting  out"  rhymes  of  Europe   and  America.     It   should 
be  added  that  the  modern  forms  given  above  are  from  Chinese  roots. 

1  Mupioi,  10000. 

2  Which  would,  if  so  considered,  appear  as  momo  chit  or  in  modern  Japa- 
nese as  hyaku  sen. 

3  So  yorozu,  a   softened    form  of  to  yorozu.     In  modern  Japanese,  jiu  man, 
man  being  the  myriad. 

4  Mille  -f-  on,  "big  thousand",  just  as  saloon  is  salle  -\-  on,  a  big  hall,  and 
gallon  is  gill  -\-  on,  a  big  gill. 

5  Momo  yorozu,  or,  in  modern  Japanese,  -hyaku  man. 

6  See,  for  example,  Gow,  J.,  History  of  Greek  Mathematics,  Cambridge  1884, 
and  similar  works. 

^  See  CoLEBROOKE,  H.  T.,  Algebra,  with  Arithmetic  and  Mensuration,  from 
the  Sanscrit  of  Brahmegupta  and  Bhascara.  London  1817,  p.  4;  TAYLOR,  J., 
Lilawati.  Bombay  1816,  p.  5. 

8  WILLIAMS,  S.  W.,  The  Middle  Kingdom.  New  York  1882;  edition  of  1895, 
vol.  I,  p.  619.  Thus  Wan  sui  is  a  myriad  of  years,  and  Wan  sui  Yeh  means 
the  Lord  of  a  Myriad  Years,  /.  e.,  the  Emperor.  The  swastika  is  used  by 
the  Buddhists  in  China  as  a  symbol  for  myriad.  This  use  of  the  myriad  in 
China  is  very  ancient. 


6  I.   The  Earliest  Period. 

convinced  that  there  must  have  been  a  considerable  intercourse 
of  scholars  at  an  early  date.1 

Of  the  rest  of  Japanese  mathematics  in  this  early  period  we 
are  wholly  ignorant,  save  that  we  know  a  little  of  the  ancient 
system  of  measures  and  that  a  calendar  existed.  How  the 
merchants  computed,  whether  the  almost  universal  finger  compu- 
tation of  ancient  peoples  had  found  its  way  so  far  to  the  East, 
what  was  known  in  the  way  of  mensuration,  how  much  of  a 
crude  primitive  observation  of  the  movements  of  the  stars  was 
carried  on,  what  part  was  played  by  the  priest  in  the  orien- 
tation of  shrines  and  temples,  what  was  the  mystic  significance 
of  certain  numbers,  what,  if  anything,  was  done  in  the  record- 
ing of  numbers  by  knotted  cords,  or  in  representing  them  by 
symbols, — all  these  things  are  looked  for  in  the  study  of  any 
primitive  mathematics,  but  they  are  looked  for  in  vain  in  the 
evidences  thus  far  at  hand  with  respect  to  the  earliest  period 
of  Japanese  history.  It  is  to  be  hoped  that  the  spirit  of  in- 
vestigation that  is  now  so  manifest  in  Japan  will  result  in 
throwing  more  light  upon  this  interesting  period  in  which 
mathematics  took  its  first  root  upon  Japanese  soil. 

*  There  is  considerable  literature  upon  this  subject,  and  it  deserves  even 
more  attention.  See,  for  example,  the  following:  KLINGSMILL,  T.  W.,  The 
Intercourse  of  China  with  Eastern  Turkestan  .  .  .  in  the  second  century  B.  C.,  in 
the  Journal  of  the  Royal  Asiatic  Society,  N.  S.,  London  1882,  vol.  XIV,  p.  74. 
A  Japanese  scholar,  T.  Kimura,  is  just  at  present  maintaining  that  his  people 
have  a  common  ancestry  with  the  races  of  the  Greco-Roman  civilization, 
basing  his  belief  upon  a  comparison  of  the  mythology  and  the  language  of 
the  two  civilizations.  See  also  P.  VON  BOHLEN,  Das  alie  Indien  mil  besonderer 
Riicksicht  anf  ^Egypten.  Konigsberg  1830;  REINAUD,  Relations politiques  et  com- 
merciales  de  F  Empire  Romain  avec  FAsie  orient  ale.  Paris  1863;  P.  A.  DI  SAN 
FlLIPO,  Delle  Relazioni  antiche  et  moderne  fra  L' Italia  e  I' India.  Rome  1 886; 
SMITH  and  KARPINSKI,  The  Hindu-Arabic  Numerals.  Boston  1911,  with  exten- 
sive bibliography  on  this  point. 


CHAPTER  II. 
The  Second  Period. 

The  second  period  in  the  history  of  Japanese  mathematics 
(552 — 1600)  corresponds  both  in  time  and  in  nature  with  the 
Dark  Ages  of  Europe.  Just  as  the  Northern  European  lands 
came  in  contact  with  the  South,  and  imbibed  some  slight 
draught  of  classical  learning,  and  then  lapsed  into  a  state  of 
indifference  except  for  the  influence  of  an  occasional  great  soul 
like  that  of  Charlemagne  or  of  certain  noble  minds  in  the 
Church,  so  Japan,  subject  to  the  same  Zeitgeist,  drank  lightly 
at  the  Chinese  fountain  and  then  lapsed  again  into  semi- 
barbarism.  Europe  had  her  Gerbert,  and  Leonardo  of  Pisa, 
and  Sacrobosco,  but  they  seem  like  isolated  beacons  in  the 
darkness  of  the  Middle  Ages;  and  in  the  same  way  Japan,  as 
we  shall  see,  had  a  few  scholars  who  tended  the  lamp  of 
learning  in  the  medieval  night,  and  who  are  known  for  their 
fidelity  rather  than  for  their  genius. 

Just  as  in  the  West  we  take  the  fall  of  Rome  (476)  and  the 
fall  of  Constantinople  (1453),  two  momentous  events,  as  con- 
venient limits  for  the  Dark  Ages,  so  in  Japan  we  may  take 
the  introduction  of  Buddhism  (552)  and  the  revival  of  learning 
(about  1600)  as  similar  limits,  at  least  in  our  study  of  the 
mathematics  of  the  country. 

It  was  in  round  numbers  a  thousand  years  after  the  death 
of  Buddha1  that  his  religion  found  its  way  into  Japan.2  The 

1  The  Shinshiu  or  "True  Sect"    of  Buddhists  place  his  death  as  early  as 
949  B.  C.,  but  the  Singalese  Buddhists   place  it   at  543  B.  C.     Rhys  Davids, 
who  has  done  so  much  to  make  Buddhism  known  to  English  readers,  gives 
412  B.  C.,  and  Max  Miiller  makes  it  477  B.  C.,  See  also  SUMNER,  J-,  Buddhism 
and  traditions  concerning  its  introduction  into    Japan,  Transactions  of  the  Asiatic 
Society  of  Japan,   Yokohama  1886,  vol.  XIV,   p.  73.     He  gives  the   death  of 
Buddha  as  544  B.  C. 

2  It  was  introduced  into  China  in  64  A.  D.,  and  into  Korea  in  372. 


8  II.  The  Second  Period. 

date  usually  assigned  to  this  introduction  is  552,  when  an  image 
of  Buddha  was  set  up  in  the  court  of  the  Mikado;  but  evidence1 
has  been  found  which  leads  to  the  belief  that  in  the  sixteenth 
year  of  Keitai  Tenno  (an  emperor  who  reigned  in  Japan  from 
507  to  531),  that  is  in  the  year  522,  a  certain  man  named 
Szu-ma  Ta2  came  from  Nan-Liang  3  in  China,  and  set  up  a 
shrine  in  the  province  of  Yamato,  and  in  it  placed  an  image 
of  Buddha,  and  began  to  expound  his  religion.  Be  this  as  it 
may,  Buddhism  secured  a  foothold  in  Japan  not  far  from  the 
traditional  date  of  552,  and  two  years  later*  Wang  Pao-san, 
a  master  of  the  calendar,  s  and  Wang  Pao-liang,  doctor  ot 
chronology,6  an  astrologer,  crossed  over  from  Korea  and  made 
known  the  Chinese  chronological  system.  A  little  later  a 
Korean  priest  named  Kanroku7  crossed  from  his  native  country 
and  presented  to  the  Empress  Suiko  a  set  of  books  upon 
astrology  and  the  calendar.8  In  the  twelfth  year  of  her  reign 
(604)  almanacs  were  first  used  in  Japan,  and  at  this  period 
Prince  Shotoku  Taishi  proved  himself  such  a  fosterer  of 
Buddhism  and  of  learning  that  his  memory  is  still  held  in  high 
esteem.  Indeed,  so  great  was  the  fame  of  Shotoku  Taishi 
that  tradition  makes  him  the  father  of  Japanese  arithmetic 
and  even  the  inventor  of  the  abacus.9  (Fig-  !•) 

A   little  later  the  Chinese  system  of  measures  was  adopted, 
and  in  general  the  influence  of  China  seems  at  once  to  have 

1  See  SUMNER,  loc.  cit.,  p.  78. 

2  In  Japanese,  Shiba  Tatsu. 

3  I.  e.,  South  Liang,  Liang  being  one  of  the  southern  monarchies. 

4  I.  e.,  in  554,  or  possibly  553. 

5  In  Europe   he  would   have  had  charge  of  the  Compotus,  the  science  of 
the  Church  calendar,  in  a  Western  monastery. 

6  Also  called  a  Doctor  of  Yih.    The  doctrine  of  Yih  (changes)  is  set  forth 
in  the  Yih  King  (Book  of  Changes),  one  of  the  ancient  Five  Classics  of  the 
Chinese.     There  is  a  very  extensive  literature  upon  this  subject. 

7  Or  Ch'iian-lo. 

8  SUMNER,  loc.  cit ,  p.  80,  gives  the  date  as  593.   Endo,  who  is  the  leading 
Japanese  authority,  gives  it  as  602. 

9  That  this  is  without  foundation  will  appear  in  Chapter  III.     The  soroban 
which  he  holds  in  the  illustration  here  given  is  an  anachronism. 


II.  The  Second  Period. 


become  very  marked.  Fortunately,  just  about  this  time,  the 
Emperor  Tenchi  (Tenji)  began  his  short  but  noteworthy  reign 
(668— 671). '  While  yet  crown  prince  this  liberal-minded  man 
invented  a  water  clock,  and  divided  the  day  into  a  hundred 
hours,  and  upon  ascending  the  throne  he 
showed  his  further  interest  by  founding  a 
school  to  which  two  doctors  of  arithmetic 
and  twenty  students  of  the  subject  were 
appointed.  An  observatory  was  also 
established,  and  from  this  time  mathema- 
tics had  recognized  standing  in  Japan. 

The  official  records  show  that  a  uni- 
versity system  was  established  by  the 
Emperor  Monbu  in  701,  and  that  mathe- 
matical studies  were  recognized  and  were 
regulated  in  the  higher  institutions  of 
learning.  Nine  Chinese  works  were  speci- 
fied, as  follows: — (i)  Chou-pei(Suan-ching), 
(2)  Sun-tsu  (Suan- eking),  (3)  Liu-chang, 
(4)  San-k'ai  Chung- ch' a,  (5)  Wu-fsao 
(Suan-s/m),  (6)  Hai-tao  (Suan-sJni), 
(7)  Chiu-szu,  (8)  Chiu-chang,  (9)  Chui- 
skn.2  Of  these  works,  apparently  the  most 
famous  of  their  time,  the  third,  fourth, 
and  seventh  are  lost.  The  others  are 
probably  known,  and  although  they  are 
not  of  native  Japanese  production  they  so  Shotoku  Taishi,  with  a 

it        .    ,,  ..     '        f  soroban.     From  a  bronze 

greatly    influenced    the    mathematics    ot 

statuette. 

Japan  as  to  deserve  some  description  at 

this    time.      We    shall   therefore    consider   them    in    the    order 

above  given. 

i.  Chou-pei  Suan-cJiing.     This  is   one    of  the   oldest  of  the 
Chinese  works    on    mathematics,  and    is    commonly  known    in 


Fig.  i. 


1  MURRAY,  D.,    The  Story  of  Japan.     N.  Y.  1894,   p.  398,  from  the  official 
records. 

2  ENDO,  Book  I,  pp.  12 — 13. 


IO  II.  The  Second  Period. 

China  as  Chow-pi,  said  to  mean  the  "Thigh  bone  of  Chow".1 
The  thigh  bone  possibly  signifies,  from  its  shape,  the  base  and 
altitude  of  a  triangle.  Chow  is  thought  to  be  the  name  of  a 
certain  scholar  who  died  in  1105  B.  C,  but  it  may  have  been 
simply  the  name  of  the  dynasty.  This  scholar  is  sometimes 
spoken  of  as  Chow  Kung,2  and  is  said  to  have  had  a  discussion 
with  a  nobleman  named  Kaou,  or  Shang  Kao,3  which  is  set 
forth  in  this  book  in  the  form  of  a  dialogue.  The  topic  is  our 
so-called  Pythagorean  theorem,  and  the  time  is  over  five  hundred 
years  before  Pythagoras  gave  what  was  probably  the  first 
scientific  proof  of  the  proposition.  The  work  relates  to  geo- 
metric measures  and  to  astronomy.4 

2.  Sun-tsu  Suan-ching.   This  treatise  consists  of  three  books, 
and   is   commonly  known  in  China   as  the  Swan-king  (Arith- 
metical   classic)    of  Sun-tsu    (Sun-tsze,   or  Swen-tse),   a    writer 
who  lived  probably  in  the  3d  century  A.  D.,  but  possibly  much 
earlier.     Ttie  work  attracted  much  attention  and  is  referred  to 
by  most   of  the  later  writers,  and  several  commentaries  have 
appeared  upon   it.     Sun-tsu   treats    of  algebraic  quantities,  and 
gives  an  example  in  indeterminate  equations.    This  problem  is 
to  "find  a  number  which,  when  divided  by  3  leaves  a  remainder 
of  2,    when    divided    by    5    leaves    3,    and    when    divided    by 
7  leaves  2."s     This  work  is   sometimes,  but  without  any  good 
reason,  assigned  to  Sun  Wu,  one  of  the  most  illustrations  men 
of  the  6th  century  B.  C. 

3.  Liu- Chang.    This  is  unknown.   There  was  a  writer  named 

'*  Pi  means  leg,  thigh,  thigh-bone. 

2  Chi  Tan,  known  as  Chow  Kung  (that  is,  the  Duke  of  Chow),  was  brother 
and  advisor  to  the  Emperor  Wu  Wang  of  the  Chow  dynasty.    It  is  possible 
that  he  wrote  the  Chow  Li,  "Institutions  of  the  Chow  Dynasty",  although  it 
is    more    probable    that    it    was    written    for    him.      The    establishment    and 
prosperity  of  the   Chow    dynasty   is   largely  due   to    him.     There  is  no  little 
doubt  as  to  the  antiquity  of  this  work,  and  the  critical  study  of  scholars  may 
eventually  place  it  much  later  than  the  traditional  date  here  given. 

3  Also  written  Shang  Kaou. 

4  For    a   translation    of    the    dialogue   see   WYLIE,   A.,    Chinese  Researches. 
Shanghai  1897,  Part  III,  p.  163. 

5  His  result  is  23.    For  his  method  of  solving  see  WYLIE,  loc.  cit.,  p.  175. 


II.  The  Second  Period.  II 

Liu  Hui1  who  wrote   a  treatise  entitled   Chung -ctt a,  but  this 
seems  to  be  No.  4  in  the  list. 

4.  San-fcai  Chung- eft  a.     This   is  also  unknown,  but   is  per- 
haps   Liu   Hui's     Cliung-cfta-keal-tsih-wang-chi-shuh     (The 
whole    system    of  measuring    by    the   observation    of  several 
beacons),  published  in  263.      The  author    also   wrote  a  com- 
mentary  on   the    Chiu-chang  (No.  8    in  this   list).     It   relates 
to  the   mensuration  of  heights   and  distances,  and   gives  only 
the  rules   without  any  explanation.     About  1250  Yang  Hway 
published  a  work  entitled  Siang-kiai-Kew-chang-Swan-fa  (Ex- 
planation of  the  arithmetic  of  the  Nine  Sections),  but  this  is  too  late 
for  our  purposes.    He  also  wrote  a  work  with  a  similar  title  Siang- 
kiai-Jeh-yung-Swan-fa  (Explanation  of  arithmetic  for  daily  use). 

5.  Wu-t'sao  Suan-shu.   The  author  and  the  date  of  this  work 
are   both   unknown,  but  it  seems  to  have  been  written  in  the 
2d    or   3d   century.2     It    is   one    of  the  standard  treatises   on 
arithmetic  of  the  Chinese. 

6.  Hai-tao  Suan-shu.   This  was  a  republication  of  No.  4,  and 
appeared  about  the  time  of  the  Japanese  decree  of  701.     The 
name  signifies  "The   Island  Arithmetical  Classic", -5  and  seems 
to  come  from  the  first  problem,  which  relates  to  the  measuring 
of  an  island  from  a  distant  point. 

7.  Chiu-szu.    This  work,  which  was  probably  a  commentary 
on  the  Suan-sJm  (Swan-king]  of  No.  8,  is  lost. 

8.  CJdu-chang.     Chiu-chang  Suan-shu*  means  "Arithmetical 
Rules  in  Nine  Sections".    It  is  the  greatest  arithmetical  classic 
of  China,   and   tradition   assigns  to   it  remote   antiquity.     It  is 
related  in  the  ancient   Tung-kien-kang-muh  (General  History  of 
China)  that  the  Emperor  Hwang-ti,s  who  lived  in  2637  B.  C, 


1  Lew-hwuy  according  to  Wylie's  transliteration,  who  also  assigns  him  to 
about  the  5th  century  B.  C. 

2  But  see  WYLIE,  loc.  cit.,  who   refers   it  to   about   the   5th   century,  and 
improperly  states  that  Wu-t'sao  is  the  author's  name.     He  gives  it  the  com- 
mon name  of  Swan  king  (Arithmetical  classic). 

3  Also  written  Hae-taou-swan-king. 

4  Kew  chang-swan-shu,  Kiu-chang-san-suh,  Kieou  chang. 

5  Or  Hoan-ti,  the  "Yellow  Emperor".  Some  writers  give  the  date  much  earlier. 


12  II.  The  Second  Period. 

caused  his  minister  Li  Show1  to  form  the  Ckiu-chang.*  Of 
the  text  of  the  original  work  we  are  not  certain,  for  the  reason 
that  during  the  Ch'in  dynasty  (220 — 205  B.  C.)  the  emperor 
Chi  Hoang-ti  decreed,  in  213  B.  C,  that  all  the  books  in 
the  empire  should  be  burned.  And  while  it  is  probable  that 
the  classics  were  all  surreptitiously  preserved,  and  while  they 
could  all  have  been  repeated  from  memory,  still  the  text  may 
have  been  more  or  less  corrupted  during  the  reign  of  this 
oriental  vandal.  The  text  as  it  comes  to  us  is  that  of  Chang 
T'sang  of  the  second  century  B.  C.,  revised  by  Ching  Ch'ou- 
ch'ang  about  a  hundred  years  later.3  Both  of  these  writers 
lived  in  the  Former  Han*  dynasty  (202  B.  C. — 24  A.  D.),  a 
period  corresponding  in  time  and  in  fact  with  the  Augustan  age 
in  Europe,  and  one  in  which  great  effort  was  made  to  restore 
the  lost  classics,  s  and  both  were  ministers  of  the  emperor. 

This  classical  work  had  such  an  effect  upon  the  mathematics 
of  Japan  that  a  summary  of  the  contents  of  the  books  or  chapters 
of  which  it  is  composed  will  not  be  out  of  place.  The  work 
contained  246  problems,  and  these  are  arranged  in  nine  sect- 
ions as  follows: 

(1)  Fang-tien,  surveying.   This  relates  to  the  mensuration  of 
various  plane  figures,  including  triangles,  quadrilaterals,  circles, 
circular  segments  and  sectors,  and  the  annulus.    It  also  contains 
some  treatment  of  fractions. 

(2)  Suh-pu  (Shu-poo).     This   treats    chiefly   of  commercial 
problems  solved  by  the  "rule  of  three". 

(3)  Shwai-fen  (Shwae-fun,  SJmai-feii).  This  deals  with  partner- 
ship. 

1  Or  Li-shou. 

2  WVLIE,  A.,    Jottings    of  the    Science    of  Chinese    Arithmetic,   North    China 
Herald  for    1852,  Shanghai  Almanac  for    1853,    Chinese  Researches,     Shanghai 
1897,  Part  III,  page   159;   BIERNATZKI,  Die  Arithmetic  der  Chinesen,   CRELLE'S 
Journal  for  1856,  vol.  52. 

3  For  this  information  we  are  indebted  to  the  testimony  of  Liu  Hui,  whose 
commentary  was  written  in  263. 

4  Also  known  as  the  Western  Han. 

5  LEGGE,  J.,  The  Chinese  Classics.     Oxford   1893,    2nd  edition,  vol.  I,  p.  4. 


II.  The  Second  Period.  13 

(4)  Shao-kang  (Sliaou-kwang).    This  relates  to  the  extraction 
of  square  and  cube  roots,  the  process  being  much  like  that  of 
the  present  time. 

(5)  Shang-kung.     This  has   reference  to  the  mensuration  of 
such    solids    as    the    prism,    cylinder,    pyramid,    circular    cone, 
frustum  of  a  cone,  tetrahedron,  and  wedge. 

(6)  Kin-sJiu  (Kiun-shoo,   Ghiin-sJni)  treats  of  alligation. 

(/)  Ying-pu-tsu  (Yung-yu,  Yin-nuk).  This  chapter  treats  of 
"Excess  and  deficiency",  and  follows  essentially  the  old  rule 
of  false  position. J 

(8)  Fang-ctieng  (Fang-cheng,  Fang- cliing).     This  chapter 
relates   to   linear   equations   involving   two    or   more   unknown 
quantities,  in  which  both  positive  (ching)  or  negative  (foo)  terms 
are  employed.     The  following  example  is  a  type:   "If  5  oxen 
and  2  sheep  cost   10  taels  of  gold,  and  2  oxen  and  8  sheep 
cost  8  taels,  what   is  the   price  of  each?"     It  is  probable  that 
this  chapter  contains  the  earliest  known  mention  of  a  negative 
quantity,    and   if  the  ancient  text  has   not  been   corrupted  Jit 
places  this  kind  of  number  between  2000  and  3000  B.  C. 

(9)  Kou-ku,  a  term  meaning  a  right  triangle.     The  essential 
feature   of  this  chapter  is  the  Pythagorean  theorem,  which  is 
stated   as  follows:   "The    first  side   and  the   second  side  being 
each    squared    and    added,    the    square    root    of    the    sum    is 
the    hypotenuse."      One    of  the  twenty -four   problems  in  this 
section  involves  the  equation  x2-  +  (20  +  14)  x  —  2  x  20  x  1775  =  o, 
and  a  rule  is  laid  down  that  is  equivalent  to  the  modern  for- 
mula for  the  quadratic.   If  these  problems  were  in  the  original 
text,  and  that  text  has  the  antiquity   usually  assigned   to   it, 
concerning   neither  of  which  we   are  at  all   certain,  then  they 
contain  the  oldest  known  quadratic  equation.   The  interrelation 
of  ancient  mathematics  is  seen  in  two  problems  in  this  chapter. 
One  is  that  of  the  reed  growing  i  foot  above  the  surface  in  the 
center  of  a  pond  10  feet  square,  which  just  reaches  the  surface 
when  drawn  to  the  edge  of  the  pond,  it  being  required  to  find  the 

1  The  Regula  falsi  or  Regula  positionis  of  the  Middle  Ages  in  Europe.    The 
rule  seems  to  have  been  of  oriental  origin. 


14  II.  The  Second  Period. 

depth  of  the  water.  The  other  is  the  problem  of  the  broken 
tree  that  has  been  a  stock  question  for  four  thousand  years. 
Both  of  these  problems  are  found  in  the  early  Hindu  works 
and  were  among  the  medieval  importations  into  Europe. 

The  value  of  it1  used  in  the  "Nine  Sections"  is  3,  as  was 
the  case  generally  in  early  times.2  Commentators  changed  this 

later,  Liu  Hui  (263)  giving  the  value  ,  which  is  equivalent  to 
3-*4-3 

9.  Chui-shu.  This  is  usually  supposed  to  be  Tsu  Ch'ung- 
chih's  work  which  has  been  lost  and  is  now  known  only  by  name. 

This  list  includes  all  of  the  important  Chinese  classics  in 
mathematics  that  had  appeared  before  it  was  made,  and  it 
shows  a  serious  attempt  to  introduce  the  best  material  available 
into  the  schools  of  Japan  at  the  opening  of  the  8th  century. 
It  seemed  that  the  country  had  entered  upon  an  era  of  great 
intellectual  prosperity,  but  it  was  like  the  period  of  Charle- 
magne, so  nearly  synchronous  with  it, — a  temporary  beacon 
in  a  dark  night.  Instead  of  leading  scholars  to  the  study  of 
pure  mathematics,  this  introduction  of  Chinese  science,  at  a 
time  when  the  people  were  not  fully  capable  of  appreciating 
it,  seemed  rather  to  foster  a  study  of  astrology,  and  mathe- 
matics degenerated  into  mere  puzzle  solving,  the  telling  of 
fortunes,  and  the  casting  of  horoscopes.  Japan  itself  was  given 
up  to  wars  and  rumors  of  wars.  The  "Nine  Sections"  was 
forgotten,  and  a  man  who  actually  knew  arithmetic  was  looked 
upon  as  a  genius.  The  samurai  or  noble  class  disdained  all 
commercial  pursuits,  and  ability  to  operate  with  numbers  was 
looked  upon  as  evidence  of  low  birth.  Professor  Nitobe  has 
given  us  a  picture  of  this  feudal  society  in  his  charming  little 
book  entitled  Bushido,  TJie  Soul  of  Japan.  *  "  Children,"  he 

1  In  Chinese  Chou-le;  in  Japanese  yenshu  ritsu. 

2  It  is  also  found  in  the  Chou-pei,  No.  1  in  this  list. 

— f    3  MlKAMI,  Y.,    On   Chinese    Circle •  Squarers,   in    the  Bibliotheca   Mathcnialica, 
1910,  vol.  X(3),  p.  193. 

4  Tokio  1905)  p.  88.  Some  historical  view  of  these  early  times  is  given 
in  an  excellent  work  by  W.H.  SHARP,  The  Educational  System  of  Japan.  Bombay 
1906,  pp.  I,  10,  II. 


II.  The  Second  Period.  15 

says,  "were  brought  up  with  utter  disregard  of  economy.  It 
was  considered  bad  taste  to  speak  of  it,  and  ignorance  of  the 
value  of  different  coins  was  a  token  of  good  breeding.  Know- 
ledge of  numbers  was  indispensable  in  the  mastering  of  forces 
as  well  as  in  the  distribution  of  benefices  and  fiefs,  but  the 
counting  of  money  was  left  to  meaner  hands."  Only  in  the 
Buddhist  temples  in  Japan,  as  in  the  Christian  church  schools 
in  Europe,  was  the  lamp  of  learning  kept  burning.  *  In  each 
case,  however,  mathematics  was  not  a  subject  that  appealed 
to  the  religious  body.  A  crude  theology,  a  purposeless  logic, 
a  feeble  literature, — these  had  some  standing;  but  mathematics 
save  for  calendar  purposes  was  ever  an  outcast  in  the  temple 
and  the  church,  save  as  it  occasionally  found  some  eccentric 
individual  to  befriend  it.  In  the  period  of  the  Ashikaga 
shoguns  it  is  asserted  that  there  hardly  could  be  found  in  all 
Japan  a  man  who  was  versed  in  the  art  of  division. 2  To  divide, 
the  merchant  resorted  to  the  process  known  as  Shokei-zan,  a 
scheme  of  multiplication  3  which  seems  in  some  way  to  have 
served  for  the  inverse  process  as  well.4  Nevertheless  the  asser- 
tion that  the  art  of  division  was  lost  during  this  era  of  constant 
wars  is  not  exact.  Manuscripts  on  the  calendar,  corresponding 
to  the  European  compotus  rolls,  and  belonging  to  the  period 
in  question,  contain  examples  of  division,  and  it  is  probable 
that  here,  as  in  the  West,  the  religious  communities  always 
had  someone  who  knew  the  rudiments  of  calendar-reckoning. 
(Fig.  2.} 

Three  names  stand  out  during  these  Dark  Ages  as  worthy 
of  mention.  The  first  is  that  of  Tenjin,  or  Michizane,  counsellor 
and  teacher  in  the  court  of  the  Emperor  Uda  (888 — 898). 

1  Notably  in  the  case  of  the  labors  of  the  learned  Kobo  Daishi,   founder 
of  the  Chenyen  sect  of  Buddhists,  who  was  born  in  774  A.  D.    See  Professor 
T.  TANIMOTO'S  address  on  Kobo  Daishi.     Kobe  1907. 

2  ENDO,  Book  I,  p.  30. 

3  UCHIDA  GOKAN,  Kokon  Sankwan,  1832,  preface. 

4  This    is    the  opinion   of  MURAI  CHIIZEN   who  lived  in   the    1 8th  century. 
See    his    Sampo    Doshi-mon.     1781.     Book  I,  article    on    the    origin   of  arith- 
metic. 


16 


II.  The  Second  Period. 


Fig.  2.     Japanese  Calendar  Rolls. 

Uda's  successor,  Daigo,  banished  him  from  the  court  and 
he  died  in  903.  He  was  a  learned  man,  and  after  his  death 
he  was  canonized  under  the  name  Tenjin  (Heavenly  man)  and 


II.  The  Second  Period. 


Fig.  3'     Tenjin,  from  an  old  bronze. 


was    looked  upon  as  the  patron  of  science  and  letters.     (See 
Fig.  3.)     The   second   is   that   of  Michinori,  Lord  of  the   pro- 
vince of  Hyuga.     His  name  is  connected  with  a  mathematical 
theory    called   the  KeisJii- 
san. I  It  seems  to  have  been 
related  to  permutations  and 
to   have   been  thought  of 
enough     consequence     to 
attract     the     attention    of 
Yoshidaa  and  of  his  great 
successor     Seki3     in     the 
1 7th  century.     Michinori's 
work    was  written    in    the 
Hogenperiod(ii56— 1159). 

The  third  name  is  that 
of  Gensho,  a  Buddhist  priest 
in  the  time  of  Shogun 
Yoriyiye,  at  the  opening 
of  the  1 3th  century.  Trad- 
ition* says  that  he  was  distinguished  for  his  arithmetical  powers, 
but  so  far  as  we  know  he  wrote  nothing  and  had  no  per- 
manent influence  upon  mathematics. 

Thus  passes  and  closes  a  period  of  a  thousand  years,  with 
not  a  single  book  of  any  merit,  and  without  advancing  the 
science  of  mathematics  a  single  pace.  Europe  was  backward 
enough,  but  Japan  was  worse.  China  was  doing  a  little,  India 
was  doing  more,  but  the  Arab  was  accomplishing  still  more 
through  his  restlessness  of  spirit  if  not  through  his  mathe- 
matical genius.  The  world's  rebirth  was  approaching,  and  this 
Renaissance  came  to  Japan  at  about  the  time  that  it  came  to 
Europe,  accompanied  in  both  cases  by  a  grafting  of  foreign 
learning  upon  native  stock. 

1  ENDd,  Book  I,  p.  28;  Murai  Chuzen,  Sampo  Doshimon, 

2  See  his  Jinko-ki  of  1627. 

3  See  Chapter  VI. 

4  See  ISOMURA  KlTTOKU,  Shusho  Ketstigisho,  1684,   Book  4,   marginal  note. 
Isomura  died  in  1710. 

2 


CHAPTER  III. 

The  Development  of  the  Soroban. 

Before  proceeding  to  a  consideration  of  the  third  period  of 
Japanese  mathematics,  approximately  the  seventeenth  century 
of  the  Christian  era,  it  becomes  necessary  to  turn  our  attention 
to  the  history  of  the  simple  but  remarkable  calculating  machine 
which  is  universal  in  all  parts  of  the  Island  Empire,  the  soroban. 
This  will  be  followed  by  a  chapter  upon  another  mechanical 
aid  known  as  the  sangi,  since  each  of  these  devices  had  a 
marked  influence  upon  higher  as  well  as  elementary  mathe- 
matics from  the  seventeenth  to  the  nineteenth  century.1 

The  numeral  systems  of  the  ancients  were  so  unsuited  to 
the  purposes  of  actual  calculation  that  probably  some  form  of 
mechanical  calculation  was  always  necessary.  This  fact  is  the 
more  evident  when  we  consider  that  convenient  writing  material 

*  The  literature  of  these  forms  of  the  abacus  is  extensive.  The  following 
are  some  of  the  most  important  sources:  VISSIERE,  A.,  Recherches  sur  I'origine 
de  fabacque  chinois,  in  Bulletin  de  Geographie.  Paris  1892;  KNOTT,  C.  G.,  The 
Abacus  in  its  historic  and  scientific  aspects,  in  the  Transactions  of  the  Asiatic 
Society  of  Japan,  Yokohama  1886,  vol.  14,  p.  18;  GOSCHKEWITSCH,  J.,  Ueber 
das  Chinesiche  Rechenbrett,  in  the  Arbeiten  der  Kaiserlich  Russischen  Gesand- 
schaft  zu  Peking,  Berlin  1858,  vol.  I,  p.  293  (no  history);  VAN  NAME,  R.,  On 
the  Abacus  of  China  and  Japan,  Journal  of  the  American  Oriental  Society,  18/5, 
vol.  X,  proc.,  p.  CX;  RODET,  L.,  Le  souan-pan  des  Chinois,  Bulletin  de  la 
Sociele  mathematique  de  France,  1880,  vol.  VIII;  DE  LA  CoUPERIE,  A.  T.,  The 
Old  Numerals,  the  Counting- Rods,  and  the  Swan-pan,  Numismatic  Chronicle, 
London  1883,  vol.  Ill  (3),  p.  297;  HAYASHI,  T.,  A  brief  history  of  Japanese 
Mathematics,  part  I,  p.  18;  HUBNER,  M.,  Die  charakteristischen  Formen  des 
Rechenbretts,  Zeitschrift  fur  Lehrmittehvesen  etc.,  Wien  1906,  II.  Jahrg.,  p.  47 
(not  historical).  There  is  also  an  extensive  literature  relating  to  other  forms 
of  the  abacus. 


III.  The  Development  of  the  Soroban.  19 

was  a  late  product,  papyrus  being  unknown  in  Greece  for 
example  before  the  seventh  century  B.  C.,  parchment  being  an 
invention  of  the  fifth r  century  B.  C,  paper  being  a  relatively 
late  product,2  and  metal  and  stone  being  the  common  media 
for  the  transmission  of  written  knowledge  in  the  earlier  centuries 
in  China.  On  account  of  the  crude  numeral  systems  of  the 
ancients  and  the  scarcity  of  convenient  writing  material,  there 
were  invented  in  very  early  times  various  forms  of  the  abacus, 
and  this  instrumental  arithmetic  did  not  give  way  to  the 
graphical  in  western  Europe  until  well  into  the  Renaissance 
period.  3  In  eastern  Europe  it  never  has  been  replaced,  for  the 
tschotii  is  used  everywhere  in  Russia  today,  and  when  one 
passes  over  into  Persia  the  same  type  of  abacus*  is  common 
in  all  the  bazaars.  In  China  the  swan-pan  is  universally  used 
for  purposes  of  computation,  and  in  Japan  the  soroban  is  as 
strongly  entrenched  as  it  was  before  the  invasion  of  western 
ideas. 

The  Japanese  soroban  is  a  comparatively  recent  invention,  having 
been  derived  from  the  Chinese  swan-pan  (Fig.  10),  which  is  also 
relatively  modern.  The  earlier  means  employed  in  China  are 
known  to  us  chiefly  through  the  masterly  work  of  Mei  Wen- 
ting  (1633 — 1721)5  entitled  Kou-swan-K>i-k'ao.6  Mei  Wen-ting 
was  one  of  the  greatest  Chinese  mathematicians,  the  author 
of  upwards  of  eighty  works  or  memoirs,  and  one  of  the  lead- 
ing writers  on  the  history  of  mathematics  among  his  people. 
He  tells  us  that  the  early  instrument  of  calculation  was  a  set. 


1  Pliny  says  of  the  second  century  B.  C. 

2  It  seems  to  have  been  brought  into  Europe  by  the  Moors  in  the  twelfth 
century. 

3  See  SMITH,  D.  E.,  Rara  Arithmetica,  Boston  1909,   index  under  Counters. 

4  Known  in  Armenia  as  the  choreb,  in  Turkey  as  the  coulba. 

5  Surnamed  Ting-kieou  and  Wou-ngan.     He  lived. in  the  brilliant  reign  of 
Kang-hi,  who   had   been   educated   partly  under   tbe   influence   of  the   Jesuit 
missionaries. 

6  Researches   on   ancient    calculating   instruments.     See  VisslEre,   loc.  cit., 
p.  7,  from  whom  I  have  freely  quoted;  WYLIE,  A.,  Notes  on  Chinese  Literature, 
p.  91. 

2* 


2O  III.  The  Development  of  the  Soroban. 

of  rods,  ch'eouS  The  earliest  definite  information  that  we  have 
of  the  use  of  these  rods  is  in  the  Han  Sim  (Records  of  the 
Han  Dynasty),  which  was  written  by  Pan  Ku  of  the  Later  Han 
period,  in  the  year  80  of  our  era.  According  to  him  the 
ancient  arithmeticians  used  comparatively  long  rods,2  and  the 
commentary  of  Sou. Lin  on  the  Han  history  tells  us  that  two 
hundred  seventy-one  of  these  formed  a  set.  3  Furthermore,  in 
the  Che-chouo  (Narrative  of  the  Century),  written  by  Lieou  Yi- 
k'ing  in  the  fifth  century,  it  appears  that  ivory  rods  were  used. 
We  also  find  that  the  ancient  ideograph  for  swan  (reckoning) 
is  1 1 1  J]"[ ,  a  form  that  is  manifestly  derived  from  the  rods,  and 
that  is  evidently  the  source  of  the  present  Chinese  ideograph. 
Mei  Wen-ting  says  that  it  is  impossible  to  give  the  origin  of 
these  rods,  but  he  believes  that  the  ancient  classic,  the  Yi/i-king, 
gives  evidence,  in  its  mystic  trigrams,  of  their  very  early  use. 4 
As  to  the  size  of  the  rods  in  ancient  times  we  are  not  informed, 
none  being  now  extant,  but  an  early  work  on  cooking,  the  Cliong- 
k'ouei-lou,  speaks  of  cutting  pieces  of  meat  3  inches  long,  like 
a  calculating  rod,  from  which  we  get  some  idea  of  their  length. 
As  to  the  early  Chinese  method  of  representing  numbers, 
we  have  a  description  by  Ts'ai  Ch'en,  surnamed  Kieou-fong 
(1167 — 1230),  a  philosopher  of  the  Song  dynasty.  In  his  Hong- 
fan  (Book  of  Annals)  he  gives  the  numerals  as  follows: 

i  ii  111  mi  HUM  mi. ..in-. .limn  iiiiT-Ti-ii 

123456789  12  25         46  69  99 


*  There  is  not  space  in  this  work  to  enter  into  a  discussion  of  the  possible 
earlier  use  of  knotted  cords,  a  primitive  system  in  many  parts  of  the  world. 
Lao-tze,  "the  old  philosopher",  refers  to  them  in  his  Tao-teh-king,  a  famous 
classic  of  the  sixth  century  B.  C.,  saying:  "Let  the  people  return  to  knotted 
cords  (chieng-shing)  and  use  them."  See  the  English  edition  by  Dr.  P.  CARCS. 
Chicago,  1898,  pp.  137,  272,  323. 

2  The  text  says  6  units  (inches)    but  we   do  not  know^the    length  of   the 
unit  (inch)  of  that  periojd. 

3  The  old  word  means,  possibly,  a  handful. 

4  The  date  of  the  Yih-King  or  Book  of  Changes  is  uncertain.     It  is  often 
spoken  of  as  Antiquissimus  Sinarum  liber,  as   in   an   edition  by  JULIUS  MOHL, 
Stuttgart,    1834 — 9,   2  vols.     It   is  ascribed  to  Fuh-hi  (B.  C.  3322)  the  fabled 
founder  of  the  nation.     There  is  an  extensive  literature  upon  the  subject. 


III.  The  Development  of  the  Soroban.  21 

Furthermore  the  great  astronomer  and  engineer  of  the  Mongol 
dynasty,  Kouo  Sheou-kin  (1281),  in  his  SJieou-she  Li,  a  treatise 
on  the  calendar,  gives  the  number  198617  in  the  following 
form,  which  may  be  compared  with  the  Japanese  sangi  of 

which  we  shall  presently  speak:  |  ||||  i  ~|~ —  J[.  This  plan 
is  much  older  than  the  thirteenth  century,  however,  for  in  the 
Snn-tsu  Snan-cJiing  mentioned  in  Chapter  II,  written  by  Sun- 
tsu  about  the  third  century,  it  is  stated  that  the  units  should 
be  vertical,  the  tens  horizontal,  the  hundreds  vertical,  the 
thousands  horizontal,  and  so  on,  and  that  for  6  one  should  not 
use  six  rods,  since  a  single  rod  suffices  for  5.  These  rules  are 
repeated,  almost  verbatim,  in  the  Hia-heou  Yang  Suan-ching, 
one  of  the  Chinese  mathematical  classics,  probably  of  the  sixth 
century.  The  rods  are  therefore  very  old,  and  they  were  the 
common  means  of  representing  numbers  in  China,  as  we  shall 
see  was  also  the  case  in  Japan,  until  a  relatively  late  period. 
As  to  the  methods  of  operating  with  the  rods,  Yang  Houei, 
in  his  Siu-kou-CJiai-ki-Swan-fa  of  1275  or  1276,  gives  the 
following  example  in  multiplication: 

=  1 1 1 1  _±   =  multiplier       =        247 
_L  1 1 1  J_  =  multiplicand  =  736 

I  J=  I  J=  ITTT  =  =  P^duct  =  181  792 
From  China  the  calculating  rods  passed  to  Korea  where  the 
natives  use  them  even  to  this  day.  These  sticks  are  commonly 
made  of  bamboo,  split  into  square  prisms,  and  numbering 
about  1 50  in  a  set.  They  are  kept  in  a  bamboo  case,  although 
some  are  made  of  bone  and  are  kept  in  a  cloth  bag  as  shown 
in  the  illustration,  (Fig.  4.).  The  Korean  represents  his  numbers 
from  left  to  right,  laying  the  rods  as  follows: 

i  ii  111  mi  x  xi  xn  xui  xini  —  T1 

123         4         56        7  8-          9          10    ii 


i  We  are  indebted  to  an  educated  Korean,  Mr.  C.  Cho,  of  the  Methodist 
Publishing  House  in  Tokio,  for  this  information.  On  the  mathematics  of 
Korea  in  general,  see  LOWELL,  P.,  The  Land  of  the  Morning  Calm.  Boston 
1886,  p.  250.  One  of  the  leading  classics  of  the  country  is  the  Song-  yang 
Jwei  soan  fa,  or  Song  yang-  houi  san  pep  (Treatise  on  Arithmetic  by  Yang  Hoei 


22 


III.  The  Development  of  the  Soroban. 


r" 


Fig.  4.     Korean  computing  rods. 


of  the  Song  Dynasty),  written  in  1275  by  Yang  Hoei,  whose  literary  name  was 
Khien  Koang;  see  M.  COURANT,  Bibliographic  Coreenne.    Paris  1896,  vol.  Ill,  p.  I. 


III.  The  Development  of  the  Soroban.  23 

The  date  of  the  introduction  of  the  rods  into  Japan  is  un- 
known, but  at  any  rate  from  the  time  of  the  Empress  Suiko 
(593 — 628  A.  D.)1  the  chikusaku  (bamboo  rods)  were  used. 
These  were  thin  round  sticks  about  2  mm.  in  diameter  and 
1 2  cm.  in  length,  but  because  of  their  liability  to  roll  they  were 
in  due  time  replaced  by  the  sangi  pieces,  square  prisms  about 
7  mm.  thick  and  5  cm.  long.  (Fig.  5.)  When  this  transition 


Fig.  5.     The  sangi  or  computing  rods.     Nineteenth  century  specimens. 

took  place   is  unknown,  nor   is  it  material  since  the  methods 
of  using  the  two  were  the  same.* 

The  method  of  representing  the  numbers  by  means  of  the 
sangi  was  the  same  as  the  one  already  described  as  having 
long  been  used  by  the  Chinese.  The  units,  hundreds,  ten 

1  HAYASHI,  T ,    A   brief  history  of  the  Japanese  Mathematics,  in  the  Nienw 
Archief  voor  Wiskunde,  tweede  Reeks,    zesde   en   sevende  Deel,  part  I,  p.   1 8. 

2  Indeed  it  is  not  certain  that  there   was   a   sudden    change   from  one  to 
the  other  or  that  the  names  signified  two  different  forms.     The  old  Chinese 
names  were   ch'eou   (which  is   the   Japanese   sangi)   and   t'se,  and   these   were 
used  as  synonymous. 


24  III.  The  Development  of  the  Soroban. 

thousands,  and  so  on  for  the  odd  places,  were  represented  as 
follows: 

I     II     III     MM     HIM     T     M"   "HT    MM 

1234       56789 

The   tens,  thousands,  hundred    thousands,   and   so    on  for  the 
the  even  places,  were  represented  as  follows: 


IO  20  30  40  50          60          70         80         90 

These  numerals  were  arranged  in  a  series  of  squares  resembling 
our  chess-board,  called  a  swan-pan,  although  not  at  all  like 
the  Chinese  abacus  that  bears  this  name.  The  following  illustra- 
tion (Fig.  6),  taken  from  Sato  Shigeharu's  Tengen  Sliinan  of 
1698,  shows  its  general  form: 


t 


Fig.  6.     The  general  form  of  the  sangi  board,  from  a  work  of  1698. 


III.  The  Development  of  the  Soroban.  25 

The  number  38057,  for  example,  would  be  represented  thus: 


III 

= 

^E 

1 

The  number  1267,  represented  by  the  sangi  without  the  ruled 
board.     Is  shown  in  Fig.  7. 

From    representing   the    numbers   by  the  sangi  on    a  ruled 
board  came  a  much  later  method  of  transferring  the  lines  to 


Fig.  7.     The  number  1267  represented  by  sattgi. 

paper,  and  using  a  circle  to  represent  the  vacant  square.  This 
could  only  have  occurred  after  the  zero  had  reached  China 
and  had  been  passed  on  to  Japan,  but  the  date  is  only  a 
matter  of  conjecture.  By  this  method,  instead  of  having  38057 
represented  as  shown  above,  we  should  have  it  written  thus: 


In  laying  down  the  rods  a  red  piece  indicated  a  positive  number 
and  a  black  one  a  negative.  In  writing,  however,  a  mark 
placed  obliquely  across  a  number  indicated  subtraction.  Thus, 

pU    meant  —  3,   and  T"    meant  —  6. 

The  use  of  the  sangi  in  the  fundamental  operations  may  be 
illustrated  by  the  following  example  in  which  we  are  required 


26 


III.  The  Development  of  the  Soroban. 


to  find  the  sho  (quotient)  given  the  jitsu  (dividend)  276,  and 
the  ho  (divisor)  I2.1 


sho 
jitsu  (276) 
ho        (12) 

II 

_L 

T 

— 

II 

First  consider  the  jitsu  as  negative,  indicating  the  fact  in  this 
manner: 


sho 
jitsu 
ho 

•H 

± 

T 

— 

II 

The  first  figure 

of  the  sho  is  evidently  2: 

— 

sho 

•H 

±. 

T 

jitsu 

— 

II 

ho 

Multiply  the  ho  by  20,  and  put  the  product,  240,  beside  the 
jitsu,  thus: 


— 

n 

±E 

T 

— 

II 

sho 

jitsu 

ho 


1  These  examples  are  taken  from  HAYASHI'S  History. 


III.  The  Development  of  the  Soroban. 
which,  by  combining  numbers  in  the  jitsu,  reduces  to 


27 


— 

sho 

$ 

T 

jitsu 

— 

II 

ho 

The  ho  is  now  advanced  one  place,  exactly  as  was  done  in 
the  early  European  plan  of  division  by  the  galley  method, 
after  which  the  next  figure  of  the  slid  is  evidently  3,  and  the 
work  appears  as  follows: 


— 

Ill 

sho 
jitsu 
ho 

^ 

T 

— 

II 

Multiplying  the  ho  by  3  the  product,  36,  is  again  written  beside 
the  jitsu,  giving 


II 


jitsu 


ho 


or 


sho 


a  result  which  is  written  thus :  1 1 1    . 

In  order  that  the  appearance  of  the  sangi  in  actual  use  may 
be  more  clearly  seen,  a  page  from  Nishiwaki  Richyu's  Sampo 
Tengen  Roku  of  1714  is  reproduced  in  Fig.  8,  and  an  illustra- 
tion from  Miyake  Kenryu's  Shojutsu  Sangaku  Zuye  of  1795  in    i/ 
Fig.  9. 


28  III.  The  Development  of  the  Soroban. 

t,    - 


y 


i  —  ^^ 

a 

y 

• 

III 

__— 

i 

111 

1 

T 

A 

* 



— 

T 

"  r^L 

*. 



II 

1 

f 

r 

« 

III 

— 

II 

0 

A  E3  7   * 
1  ?£* 

I§]  t  IP)   V   ?  ^  3%  %_ 

m  ^  ^  .  T  ^  '^^l 

L  -^  ife  El    H  ^  - 

a^  >»  \         •  .*   >    —    ^ 

Fig.  8.     Sangi  board.     From  Nishiwaki  Richyu's  Sampo  Tettgen  Roku  of  1 7 14. 

In  the  later  years  of  the  sangi  computation  the  custom  of 
arranging  the  even  places  differently  from  the  odd  places 
changed,  and  instead  of  representing  38057  by  the  old  method1 
as  shown 'on  page  25,  it  was  represented  thus: 

1  Called  Son-shi-Reppu-ho,  the  Method  of  arrangement  of  Sun-tsu. 


III.  The  Development  of  the  Soroban. 


III  TIT 


TT 


This  was  done  only  on  the  ruled  squares,  however,  the  written 
form  remaining  as  shown  on  page  25. 

The  transition  from  the  cJteou  or  rod  calculation  to  the 
present  form  of  abacus  in  China  next  demands  our  attention. 
Mei  Wen-ting,  whose  name  has  already  been  mentioned,  ex- 
presses regret  that  an  exact  date  for  the  abacus  cannot  be 


Fig.  9.     From  Miyake  Kenryu's  work  of  1795. 

fixed.  He  says,  however,  <llf,  in  my  ignorance,  I  may  be 
allowed  to  hazard  a  guess,  I  should  say  that  it  began  with  the 
first  years  of  the  Ming  Dynasty."  This  would  be 
when  T'ai-tsou,  the  first  Ming  emperor,  undertook  to  refo 
the  calendar.  At  any  rate,  Mei  Wen-ting  concludes  that  in 
the  reform  of  the  calendar  in  1281  rods  were  used,  while  in 
that  of  1384  the  abacus  was  employed.  There  is  evidence, 
however,  that  the  abacus  was  known  in  China  in  the  twelfth 
century,  but  that  it  was  not  until  the  fourteenth  that  it  was 
commonly  used.1  Since  a  division  table  such  as  is  used  in 
manipulating  the  swan-pan  is  given  in  a  work  by  Yang  Hui 
who  flourished  at  the  close  of  the  Song  Dynasty,  in  the  latter 

1  VISSIKRE,  loc.  dt.\  MIKAMI,  Y.,  A  Remark  on  the  Chinese  Mathematics  in 
Cantor's  Geschichtc  der  Mathemalik,  Archiv  dcr  Mathematik  und Physik,  vol.  XV  (3), 
Heft  i. 


30  III.  The  Development  of  the  Soroban. 

half  of  the  thirteenth  century,  we  have  reason  to  believe  that 
the  swan-pan  was  known  at  that  time.  Moreover  we  have  the 
titles  of  several  books  such  as  Chon-pan  Chi  and  Pan-chou  CJd 
recorded  in  the  Historical  Records  of  the  Song  Dynasty,  which 
seem  to  refer  to  this  instrument.  It  must  also  be  admitted 
that  at  least  one  much  earlier  work  mentions  "computations 
by  means  of  balls,"  although  this  seems  to  have  been  only  a 


Fig.  10.     The  Chinese  swan-pan,  indicating  the  number  27091. 

local  plan  known  to  but  few.  That  the  Roman  abacus  should 
have  been  known  very  early  in  China  is  not  only  probable 
but  fairly  certain,  in  view  of  the  relations  between  China  and 
Italy  at  the  time  of  the  Caesars.1 

The  Chinese  abacus  is  known  commonly  as  the  swan-pan 
(swan -/an,  "reckoning  table").  In  southern  China  it  is  also 
known  as  the  soo-pan,z  and  in  Calcutta,  where  the  Chinese 
shroffs  employ  it,  the  name  is  corrupted  to  swinbon.  The 
literary  name  is  cliou-p'an  ("ball  table"  or  "pearl  table").  As 
will  be  seen  by  the  illustration  there  are  five  balls  below  the 

1  See  SMITH  and  KARPINSKI,  loc.  tit.,  p.  79. 

2  BOWRING,  J.,  The  Decimal  System.     London  1854,  p.  193. 


III.  The  Development  of  the  Soroban. 


s- 


line  and  two  above,  each  of  the  latter  counting  as  five.  In 
the  illustration  (Fig.  10)  the  balls  are  placed  to  represent  27091. 
The  balls  are  called  chou  (pearls)  or 
tse  (son,  child,  grain),  and  are  common- 
ly spoken  of  as  swan- fan  chon-tse. 
The  transverse  bar  is  the  leang  (beam) 
or  tsi-leang  (spinal  colum,  also  used 
to  designate  the  ridge-pole  of  a  roof). 
The  columns  are  called  wei  (positions), 
hang  (lines),  or  tang  (steps,  or  bars). 
The  left  side  is  called  ts'ien  (front) 
and  the  right  side  heou  (rear).  This 
was  the  instrument  that  replaced  the 
ancient  rods  about  the  year  1300,  per- 
haps suggested  by  the  ancient  Roman 
abacus  which  it  resembles  quite  closely, 
perhaps  by  some  form  of  instrument 
in  Central  Asia,  and  perhaps  invented 
by  the  Chinese  themselves.  The  re- 
semblance to  the  Roman  form,  and 
the  known  intercourse  with  the  West, 
both  favor  the  first  of  these  hypo- 
theses. 

Just  as  the  Japanese  received  the 
sangi  from  China,  perhaps  by  way  of 
Korea,  so  they  received  the  abacus 
from  the  same  source.  They  call  their 
nstrument  by  the  name  soroban,  which 
some  have  thought  to  be  a  corruption 
of  the  Chinese  swan-pan, T  and  others 
to  have  been  derived  from  the  word 
soroiban,  meaning  an  orderly  arranged 
table. 2 

The  soroban  is  an  improvement  upon 
the  swan-pan,  as  will  be  seen  by  the  illustration.  Instead  of 

1  KNOTT,  loc.  cit.,  p.  45. 

2  OYAMADA,  Matsunoya  Hikki. 


?2  III.  The  Development  of  the  Soroban. 

having  two  5-balls  it  has  only  one,  and  it  replaces  the  balls 
by  buttons  having  a  sharp  edge  that  the  finger  easily  engages 
without  slipping.  In  the  illustration  (Fig.  n)  the  number  90278 
is  represented  in  the  center  of  the  soroban. 

The  invention  of  the  soroban,  or  rather  the  importation  and 
the  improvement  of  the  swan-pan,  is  usually  assigned  to  the 
close  of  the  sixteenth  century,  although  we  shall  show  that 
this  is  probably  too  late  a  date.  In  the  Sampo  Tamatebako, 
by  Fukuda  Riken,  published  in  1879,  an  account  is  given  of 
the  journey  of  one  Mori  Kambei  Shigeyoshi,  a  scholar  of  the 
sixteenth  century,  to  China.  Mori  was  in  his  early  days  in 
the  service  of  Lord  Ikeda  Terumasa,  and  was  afterwards  a 
retainer  of  the  great  hero  Toyotomi  Hideyoshi,  better  known 
as  Taiko,  who  in  the  turbulent  days  of  the  close  of  the  Ashi- 
kaga  Shogunate1  subdued  the  entire  country,  compelling  peace 
by  force  of  arms.  The  story  goes  that  Taiko,  wishing  to 
make  his  court  a  center  of  learning,  sent  Mori  to  China  to 
acquire  the  mathematical  knowledge  that  was  wholly  wanting 
in  Japan  at  that  period.  Mori,  however,  was  a  man  of  humble 
station,  and  his  requests  on  behalf  of  his  master  were  treated 
with  such  contempt  that  he  returned  to  his  native  land  with 
little  to  show  for  his  efforts.  Upon  relating  his  trials  and 
humiliation  to  Taiko,  the  latter  bestowed  upon  him  the  title  of 
Dewa  no  Kami,  or  Lord  of  Dewa.  Again  Mori  set  out  for 
China,  but  again  he  was  destined  to  meet  with  some  dissap- 
pointment,  for  hardly  had  he  set  foot  on  Chinese  soil  than 
Taiko  began  his  invasion  of  Korea.  China  at  once  became 
involved  in  the  defence  of  what  was  practically  a  vassal  state, 
and  as  the  war  progressed  it  became  more  and  more  a  matter 
of  danger  for  a  Japanese  to  reside  within  her  borders.  Mori 
was  not  received  with  the  favor  that  he  had  hoped  for,  and 
in  due  time  returned  to  his  native  land.  Although  he  had  spent 
some  time  abroad,  he  had  not  accomplished  his  entire  purpose. 
Nevertheless  he  brought  back  with  him  a  considerable  knowledge 


1  This  just   preceded    the   Tokugawa   shognnate,  which   lasted    from  1603 
to  1868. 


III.  The  Development  of  the  Soroban.  33 

of  Chinese  mathematics,  and  also  the  swan -pan,  which  was 
forthwith  developed  into  the  present  soroban.  If  the  story  is 
true,  Mori  must  have  spent  some  years  in  China,  for  Taiko 
began  his  invasion  in  1592  and  died  in  1598,  and  he  was 
already  dead  when  Mori  returned.  Mori  repaired  to  the  Castle 
of  Osaka  which  Taiko  had  built  and  where  he  had  lived,  and 
there  he  was  hospitably  received  by  the  son  and  successor 
of  the  great  warrior.  There  he  lived  and  wrote  until  the 
city  was  besieged  in  1615,  and  the  castle  taken  by  Japan's 
greatest  hero,  Tokugawa  lyeyasu,  founder  of  the  Tokugawa 
shogunate,  whose  tomb  at  Nikko  is  a  Mecca  for  all  tourists 
to  that  delightful  region.  We  are  told  by  Araki,1  who  lived 
at  the  beginning  of  the  eighteenth  century,  that  Mori  thence- 
forth taught  the  soroban  arithmetic  in  Kyoto. 

Although  this  story  of  Mori's  visit  to  China  and  of  his  intro- 
duction of  the  soroban  is  a  recent  one,  it  has  been  credited 
by  some  of  the  best  writers  in  Japan.2  Nevertheless  there  is 
a  good  deal  of  uncertainty  about  his  journey,3  and  still  more 
about  his  having  been  the  one  to  introduce  the  soroban  into 
Japan.  Fukuda  Riken  who,  as  we  have  said,  first  published 
the  story  in  1879,  gives  no  sources  for  his  information.  He 
received  his  information  largely  from  his  friend  C.  Kawakita, 
who  tells  the  writers  that  it  was  Uchida  Gokan  who  started 
the  story  of  Mori's  first  Chinese  journey,  claiming  that  he  had 
read  it  once  upon  a  time  in  a  certain  old  manuscript  that 
was  in  the  library  of  Yushima,  in  Yedo.  Unfortunately  on 
the  dissolution  of  the  shogunate,  at  the  time  of  the  rise  of 

1  In  the  Araki  Son-yei  Chadan,  or   Stories   told  by  Araki  (Hikoshiro)  Son- 
yei  (1640—1718). 

2  ENDO,  Book  I,  p.  45 — 46,  54—56;  HAYASHI,  History,  p.  30,  and  his  bio- 
graphical sketch  of  Seki  Kowa   in   the   Honcho  Siigaku  Koenshii   (Lectures  on 
the  Mathematics  of  Japan),  1908,  pp.  8 — to. 

3  For  example,   ALFRED  WESTPHAL   claims   that  it  was  Korea  rather  than 
China  that  Mori  visited.     See   his   Beitrag   zur   Geschichte  der  Mathematik,  in 
the    Mittheilungen    der    deutschen    Gesellschaft  fur  Natur-   iind   Volkerkunde    Osl- 
asiens  in  Tokyo,  IX.  Heft,  1876.     The  Chinese  journey  is  looked  upon  as  fic- 
tion by  the   learned    C.  Kawakita,  who   has   studied    very   carefully  the    bio- 
graphies of  the  Japanese  mathematicians. 

3 


34  HI.  The  Development  of  the  Soroban. 

the  modern  Empire,  the  books  of  this  library  were  dispersed 
and  the  manuscript  in  question  seems  to  have  been  irretrievably 
lost.  That  Uchida  claims  to  have  seen  it  we  have  been  per- 
sonally informed  both  by  Mr.  Kawakita  and  by  Mr.  N.  Oka- 
moto,  to  whom  he  told  the  circumstance.  Nevertheless  as 
historical  evidence  all  this  is  practically  worthless.  Uchida  was 
a  learned  man,  but  his  reputation  was  not  above  reproach. 
He  never  told  the  story  until  the  manuscript  had  disappeared, 
and  no  one  has  the  slightest  idea  of  the  age,  the  character, 
or  the  reliability  of  the  document.  Moreover  the  older  writers 
make  no  mention  of  this  Chinese  journey,  as  witness  the  Araki 
Son-yei  Chadan  which  was  written  only  a  century  after  Mori 
lived  and  which  gives  a  sketch  of  his  life  and  a  brief  state- 
ment concerning  the  early  Japanese  mathematics.  In  Murai's 
Sampo  Doshi-mon, *  written  nearly  a  century  later  still,  no  men- 
tion is  made  of  the  matter.  Indeed,  it  is  not  until  after  the 
story  was  started  by  Uchida  that  we  ever  hear  of  it.2 

But  whether  or  not  Mori  went  to  China,  he  did  much  for 
mathematics  and  he  was  an  expert  in  the  manipulation  of 
the  soroban.  He  was  also  possessed  of  a  well-known  Chinese 
treatise  on  the  swan-pan,  written  by  Ch'eng  Tai-wei3  and 
published  in  I593,4  a  work  that  greatly  influenced  Japanese 
mathematics  even  long  after  Mori's  death.  Mori  himself  publish- 
ed a  work  on  arithmetic  in  two  books  entitled  Kijo  Ranjd5, 
and  he  left  a  manuscript  on  mathematics  written  in  1628. 6 
Both  have  been  lost,  however,  and  of  the  contents  of  neither 

1  Book  I,  chapter  on  the  Origin  of  Arithmetic,  published  in  1781. 

*  The  oldest  manuscript  that  we  have  found  that  speaks  of  it  is  SHIRAISHI'S 
Siika  Jimmei-Shi,  but  since  the  author  was  a  contemporary  of  Uchida  he 
probably  simply  related  the  latter's  story. 

3  Erroneously  given  in  ENDO  as  Ju  Szu-pu.     Book  I,  p.  45. 

4  The  Suan-fa  Tung-tsong. 

5  The  Kijoho   method  of  division  on  the   soroban,  described  later.     See 
MURAI,  Sampo  Doshi-mon,  1781,  Book  I;  and  ENDO,  Book  I,  p.  45. 

6  This   fact  is  recorded  in  an  anonymous  manuscript  entitled  Sanwa  Zni- 
hitsu,  which  relates  that  the  original  manuscript,  signed    and  sealed  by  Mori 
himself,  was  in  the  possession  of  a  mathematician  named  Kubodera  early  in 
the  nineteenth  century. 


III.  The  Development  of  the  Soroban.  35 

have  we  any  knowledge.  Mori  seems  to  have  made  a  livelihood 
after  the  fall  of  Osaka  by  teaching  arithmetic  in  Kyoto,  where 
hundreds  of  pupils  flocked  to  learn  of  him  and  study  with  the 
man  who  proclaimed  himself  "The  first  instructor  in  division 
in  the  world."  He  is  said  to  have  spent  his  last  years  at  Yedo, 
the  modern  Tokyo.  Three  of  his  pupils,1  Yoshida  Koyu,  Ima- 
mura  Chisho,  and  Takahara  Kisshu,  known  to  their  contempo- 
raries as  "The  three  Arithmeticians,"2  did  much  to  revive  the 
study  of  the  science  in  what  we  have  designated  as  the  third 
period  of  Japanese  mathematics,  and  of  them  we  shall  speak 
more  at  length  in  a  later  chapter. 

There  are  various  reasons  for  believing  that  the  swan-pan 
was  not  first  brought  to  Japan  by  Mori.  In  the  first  place, 
such  simple  devices  of  the  merchant  class  usually  find  their 
way  through  the  needs  of  trade  rather  than  through  the  efforts 
of  the  scholar.  It  was  so  with  the  Hindu- Arabic  numerals  in 
the  West, 3  and  it  was  probably  so  with  the  swan- pan  in  the 
East.  There  is  a  tradition  that  another  Mori,4  Mori  Misaburo, 
an  inhabitant  of  Yamada  in  the  province  of  Ise,  owned  a  swan- 
pan  in  the  Bun-an  Era,  i.  e.,  in  1444-1449.  This  instrument 
is  still  preserved  and  is  now  in  the  possession  of  the  Kita- 
batake  family,  s  It  is  also  related  that  the  great  general  and 
statesman  Hosokawa  Yusai,  in  the  time  of  Taiko,  owned  a 
small  ivory  soroban,  but  of  course  this  may  have  come  from 
his  contemporary  Mori  Kambei.  It  is,  however,  reasonable  to 
believe  that,  with  the  prosperous  intercourse  between  China 
and  Japan  during  the  Ashikaga  Shogunate,  from  the  fourteenth  to 
the  end  of  the  sixteenth  centuries  the  swan-pan  could  not  have 
failed  to  become  known  to  the  Japanese  merchants,  even  if  it 
was  not  extensively  used  by  them.  On  the  other  hand,  Mori 
Kambei  was  the  first  great  teacher  of  the  art  of  manipulating  it, 

1  See  ENDO,  Book  I,  p.  55,  and  the  Araki  Son-yei  Chadan. 

2  Also  as  the  San-shi,  or  "three  honorable  scholars." 

3  See  SMITH  and  KARPINSKI,  be.  df.t  p.  114. 

4  Not  Mori,  however. 

5  It  was  exhibited  not  long  ago  in  Tokyo.     We  are  indebted  for  this  in- 
formation to  Mr.  N.  OKAMOTO. 

3* 


III.  The  Development  of  the  Soroban. 


so  that  much  credit  is  due  to  him  for  its  general  adoption.    We 
may,  therefore,  fix  upon  about  the  year  1600  as  the  beginning 

of  the  use  of  the  soroban,  and  the 
century  from  1600  to  1700  as  the 
period  in  which  it  replaced  the  ancient 
bamboo  rods. 

It  is  proper  in  this  connection  to 
give  a  brief  description  of  the  soroban 
and  of  the  method  of  operating  with 
it,  particularly  with  a  view  to  the  needs 
of  the  Western  reader.  As  already 
stated,  the  value  of  the  ball  above 
the  beam  is  five,  one  being  the  value 
of  each  ball  below  the  beam.  In 
Fig.  12  the  right-hand  column  has 
been  used  to  represent  units,  the  next 
one  tens,  and  so  on.  In  the  picture 
these  columns  have  been  numbered 
by  arranging  the  balls  so  that  the 
units  are  I,  the  tens  2,  the  hundreds 
3,  and  so  on.  As  a  result,  the  number 
represented  isr98765432i.1 

To  add  two  numbers  we  have  only 
to  set  down  the  first  as  in  the  illu- 
stration and  then  set  down  the  second 
upon  it.  Thus  to  add  2  and  2,  we 
put  2  balls  at  the  top  of  the  colunn 
and  then  2  more,  making  4.  To  add 
2  and  3,  we  put  2  balls  at  the  top, 
and  then  add  3 ;  but  since  this  makes 
5  we  push  back  the  5  balls  and  move 
down  the  one  above  the  beam.  To 
add  4  and  3,  we  take  4  balls;  then 
we  add  the  3  by  first  adding  r,  moving 
down  the  one  above  the  beam  to  replace  the  5,  and  then 

1  The   best   description   of  this  instrument,   in   English,   is    that   given  by 
KNOTT,  he.  dt.,  p.  45. 


III.  The  Development  of  the  Soroban.  37 

adding  2  more,  leaving  the  five-ball  and  2  unit  balls.  To  add  7 
and  6,  we  set  down  the  7  by  moving  the  five-ball  and  2  unit 
balls;  we  then  move  3  more  balls,  which  give  us  10,  and  we 
indicate  this  by  moving  i  ball  in  tens'  column,  clearing  the 
units'  column  at  the  same  time,  and  then  we  add  3  more, 
making  i  ten  and  3  units.  It  will  be  seen  that  as  fast  as 
any  number  is  set  down  it  is  thereby  added  to  the  preceding 
sum,  thus  making  the  work  very  rapid  in  the  hands  of  a  skilled 
operator.  Subtraction  is  evidently  performed  with  equal  ease. 
For  multiplying  readily  on  the  soroban  it  is  necessary  to 
learn  the  multiplication  table.  In  this  table  the  Japanese  have 
two  points  of  advantage  over  the  Western  peoples:  (i)  they 
do  not  use  the  words  "times"  or  "equals",  thus  saving  con- 
siderably in  time  and  energy  whenever  they  employ  it;  (2)  they 
learn  their  products  only  one  way,  as  6  7's  but  not  7  6's.  Thus 
their  table  for  6  is  as  follows : z 

Japanese  names  In  our  figures 

ichi  roku  roku2  166 

ni  roku  ju  ni  2     6     12 

san  roku  3          ju  hachi  3     6     18 

shi  roku  ni  ju  shi  4     6     24 

go  roku  san  ju  5     6     30 

roku  roku  san  ju  roku  6     6     36 

roku  shichi          shi  ju  ni  67     42 

roku  hachi*         shi  ju  hachi  6     8     48 

roku  kus  go  ju  shi  6     9     54 

This  table  reminds  us  of  the  one  in  common  use  by  the 
Italian  merchants  from  the  fourteenth  to  the  sixteenth  century, 
and  which  was  probably  quite  universal  in  the  mercantile  houses. 

For  purposes  of  historic  interest  we  take  to  illustrate  the 
process  of  multiplication  an  example  from  the  Jinko-ki  of 

1  KNOTT,  loc.  at.,  p.  50. 

2  This  is  usually  stated  as  "in  roku  ga  roku"  the  ithi  being  corrupted  to  in 
and  the  ga  inserted  for  euphony. 

3  Corrupted  to  sabu  roku. 

4  The  hachi  is  abbreviated  to  ha  in  this  case,  for  euphony. 

5  Roku  ku  may  here  be  abbreviated  to  rokku. 


154988 


38  III.  The  Development  of  the  Soroban. 

Yoshida,  published  in  1627,  and  described  more  fully  in 
Chapter  V.  To  multiply  625  by  16  the  multiplier  is  placed 
to  the  left  of  the  multiplicand  on  the  soroban,  a  plan  that  is 
exactly  opposite  to  the  Chinese  arrangement  as  set  forth  in  the 
Suan-fa  Tung-tsong  of  1593.  It  represents  one  of  the  im- 


Fig.  13.     16     625. 

provements    of   Mori   or    of   Yoshida,    and    has    always    been 
followed  in  Japan. 

We  first  take  the  partial  product  5  x  6  =  30,  and  place  the 
30  just  to  the  right    of  the  625, 1  so  that   the  soroban    reads 

16     62530 


Fig.  14.     16     62530. 

We  now  take  5x1  =  5,   and  add  this  5  to  the  3,  making 
the  product  80  thus  far.     The  5  of  the  625  now  having  been 

1  In  general,  the  units'  figure  of  this  product  is  placed  as  many  columns 
to  the  right  as  there  are  figures  in  the  multiplier. 


III.  The  Development  of  the  Soroban.  39 

multiplied  by   16,  it  is  removed,  so  that  the  figures  stand  as 
follows:  16     62080 


Fig.  15.     16     62080. 

The  next  step  is  the  multiplication  of  2  by  16,  and  this  is 
done  precisely  as  the  5  was  multiplied.  Expressed  in  figures 
the  operation  on  the  soroban  is  as  follows: 

1 6     62080 
2x6=  12 

2x1=    2^ 

Cancel  2         16     60400 

the  2  in  62080  being    removed    because   the  multiplication  of 
2  by  1 6  has  been  effected. 


Fig.  1 6.     1 6     60400. 

The  next  step  is  the  multiplication  of  6  by  16,  and  the  work 
appears  on  the  soroban  as  follows: 

1 6     60400 
6x6=  36 

1x6= 6 

16     loooo 


40  III.  The  Development  of  the  Soroban. 

The  result  is  therefore  10000. 


Fig.  17.    16    i  oooo. 

The  process  of  division  is  much  more  complicated,  and  re- 
quires the  perfect  memorizing  of  a  table  technically  known  as 
the  Ku  ki  ho,  or  "Nine  Returning  Method."  It  is  given  here 
only  for  2,  6,  and  7.* 

Ni  ichi  ten  saku  no  go        21  replace  by  5 
Nitchin  in  ju2  22  gives  I  ten 

Ni  shi  shin  ga  ni  ju  2  4  gives  2  tens 

Ni  roku  shin  ga  san  ju        26  gives  3  tens 
Ni  hachi  shin  ga  shi  ju       2  8  gives  4  tens 

Table  for  6. 

Roku  ichi  kakka  no  shi  6  i    14 

Roku  ni  san  ju  no  ni  62  32 

Roku  san  ten  saku  no  go  6  3  50 
Roku  shi  roku  ju  no  shi  64  64 
Roku  go  hachi  ju  no  ni  65  82 

Roku  chin  in  ju  6  6  gives  i  ten 

Table  for  7. 

Shichi  ichi  kakka  no  san  7  i  13 
Shichi  ni  kakka  no  roku  7  2  26 

Shichi  san  shi  ju  no  ni  73  42 

Shichi  shi  go  ju  no  go  7  4  55 

Shichi  go  shichi  ju  no  ichi  7  5  71 
Shichi  roku  hachi  ju  no  shi  7  6  84 
Shichi  chin  in  ju  7  7  gives  I  ten 

1  KNOTT,  loc.  tit.,  as  corrected  by  Mr.  MIKAMI. 

2  This  and  some  others  are  given  in  the  usual  abridged  form. 


III.  The  Development  of  the  Soroban.  41 

The  table  is  not  so  unintelligible  as  it  seems  to  a  stranger, 
and  in  fact  its  use  has  certain  advantages  over  Western  me- 
thods. In  the  first  place  it  is  not  encumbered  with  such  words 
as  "divided  by"  or  "contained  in,"  and  in  the  second  place  it 
is  not  carried  beyond  the  point  where  the  dividend  number  as 
expressed  in  the  table  equals  the  divisor.  It  is  in  fact  merely 
a  table  of  quotients  and  remainders.  Consider,  for  example, 
the  table  for  7.  This  states  that 

10:7=  I,  and  3  remainder 

20  :  7  =  2,  and  6  remainder 

30  :  7  =  4,  and  2  remainder 

40  :  7  =  5,  and  5  remainder 

50 :  7  =  7,  and  I  remainder 

60 :  7  =  8,  and  4  remainder 

70 :  7  =  10 

Taking  again  an  example  from  the  classical  work  of  Yoshida, 
let  us  divide  1234  by  8.  These  numbers  will  be  represented 
on  the  soroban  in  the  usual  way,  and  placed  as  follows: 

8     1234 

The  table  now  gives  "8  I  12",  meaning  that  IO:8— I,  with 
a  remainder  2.  We  therefore  leave  the  I  untouched  and  add 
2  to  the  next  figure,  the  numbers  then  appearing  as  follows: 

8     1434 

where  the  i  represents  the  first  figure  in  the  quotient,  and  434 
represents  the  next  dividend. 

The  table  now  tells  us  "8  4  50",  meaning  that  40 :  8  =  5, 
with  no  remainder.  We  therefore  remove  the  first  4  and  put 
5  in  its  place,  the  soroban  now  indicating 

8     1534 

where  15  represents  the  first  two  figures  in  the  quotient,  and 
34  represents  the  next  dividend. 

The  table  now  tells  us  "83  36",  meaning  that  30 : 8  =  3, 
with  a  remainder  6.  This  means  that  the  next  figure  of  the 
quotient  is  3,  and  that  we  have  6  +  4  still  to  divide.  The  soroban 
is  therefore  arranged  to  indicate 

8     153  (10) 


42  III.   The  Development  of  the  Soroban. 

But  10  :  8  =  I,  with  a  remainder  2,  so  the  soroban  is  arranged 
to  indicate  8     1542 

meaning  that  the  quotient  is  1 54  and  the  remainder  is  2.  We 
may  now  consider  the  result  is  154  1/4,  or  we  may  continue 
the  process  and  obtain  a  decimal  fraction. 

If  the   divisor  has  two  or  more  figures  it  is  convenient  to 
have  the  following  table  in  addition  to  the  one  already  given: 
i  with  i,  make  it  91 


2        „ 

2,         „ 

,,  92 

3      „ 

3,      „ 

»  93 

4      „ 

4,       „ 

»  94 

5      „ 

5,      ,, 

»  95 

6      , 

6,      „ 

,  96 

7     „ 

7>      „ 

„  97 

8      „ 

8,      „ 

»  98 

9      „ 

9,      „ 

„  99 

This  means  that  10 :  I  =9  and  I  remainder,  20 : 2  =  9  and 
2  remainder,  and  so  on. 

We  shall  sketch  briefly  the  process  of  dividing  289899  by 
486  as  given  by  Yoshida.  Arrange  the  soroban  to  indicate 

486     289899. 

The  table  gives  "4  2  50",  so  we  change  the  2  to  5  and 
arrange  the  soroban  to  indicate  the  following: 

486     589899 
5x8=  40 

5x6=    30_ 

486     546899 

Here  5  is  the  first  figure  of  the  quotient  and  46899  is  the 
remainder  to  be  divided.  Looking  now  at  the  last  table  we 
find  "4  4  94",  so  we  change  the  4  to  9  and  add  4  to  the 
following  digit.  The  soroban  is  arranged  to  indicate  the  following: 

486     546899 

Then  486     596899 

Add  4  4 

Then  9x8=  72 

9x6= 54 

Subtract  72  and  54      486     593159 


III.  The  Development  of  the  Soroban. 


43 


Here   59  is  the  first  part  of  the  quotient  and  3159  is  the 
remainder  to  be  divided. 

Proceeding  in  the  same  way,  the  next  figure  in  the  quotient 
is  6,  and  the  soroban  indicates 

486     596759 
486     596243 
486    5965 
and  the  quotient  is  596.5. 


Fig.  1 8.     From  the  work  of  Fujiwara  Norikaze,  1825. 

This  method  of  division  is  that  given  in  the  Jinko-ki,  but  in 
1645  another  plan  was  suggested  by  a  well-known  teacher, 
Momokawa  Chubei. x  This  was  the  Slid j oho,  or  method  of  di- 
vision by  the  aid  of  the  ordinary  multiplication  table,  as  in 
wiitten  arithmetic.  Momokawa  sets  it  forth  in  a  work  entitled 

1  ENDO  gives  his  personal  name  as  Jihei,  but  this  is  open  to  doubt. 


44 


III.  The  Development  of  the  Soroban. 


Kamei-zan  (1645),  and  thenceforth  the  method  itself  bore  this 
name.     This  plan,  like  the  Jinkoki,  is  fundamentally  a  Chinese 


IftfliflHUsUlJ 

^^^    >N.^-C    A    ^  -£  >X—    ^   >X  ^  >Xg7J^ 


?  ^ 

^^L 


Fig.  19.     From  an  anonymous  Kwaisanki  of  the  seventeenth  century. 

method,  as  it  appears  in  the  Suan  fa  T'ung-tsong  of  1593,  but 
it  has  never  been  so  popular  in  Japan  as  the  one  given  by 
Yoshida  in  the  Jinkoki. 


III.  The  Development  of  the  Soroban. 


45 


It  is  hardly  worth  while  to  consider  the  method  of  extracting 
roots  by  the  help  of  the  soroban,  since  the  general  theory  does 
not  differ  from  the  one  used  in  the  West,  and  the  subsidiary 
operations  have  been  sufficiently  explained. 

Although  the  soroban  began  to  replace  the  bamboo  rods 
soon  after  1600,  it  took  more  than  a  century  for  the  latter  to 
disappear  as  means  for  computation,  and,  as  we  shall  see,  they 
continued  to  be  used  for  about  two  hundred  years  longer  in 
connection  with  algebraic  work.  In  Isomura  Kittoku's  Sampo 
Ketsugi-sho  of  1660  (second  edition  1684),  and  Sawaguchi's 
Kokon  Sampo-ki  of  1670,  for  example,  we  find  both  the  rods 


Fig.  20.     From  Miyake  Kenryu's  work  of  1795- 

and  the  soroban  explained,  and  in  another  work  of  1693  only 
the  rods  are  given.  The  Tengen  Shinan,  by  Sato  Shigeharu, 
printed  in  1698,  also  gives  only  the  rods,  as  does  the  Kwatsuyo 
Sampo  (Method  of  Mathematics)  which  Araki  Hikoshiro  Son- 
yei,  being  old,  caused  his  pupil  Otaka  Yoshimasa  to  prepare 
in  1709.*  In  Murata  Tsushin's  Wakan  Sampo,  published  in 
1743,  both  systems  are  used,  and  in  a  primary  arithmetic 
printed  in  1781  only  the  rods  are  employed,  so  that  we  see 
that  it  was  a  long  time  before  the  soroban  completely  replaced 
the  more  ancient  method  of  computation.  In  general  we  may 
say  that  all  algebras  used  the  sangi  in  connection  with  the 
"celestial  element"  method  of  solving  equations,  explained  in 
the  next  chapter,  while  little  by  little  the  soroban  replaced  them 

1  It  was  printed  in  1712. 


46  III.  The  Development  of  the  Soroban. 

for  arithmetical  work.  The  pictures  of  children  learning  to  use 
the  soroban  are  often  interesting,  as  in  the  one  from  the  arith- 
metic of  Fujiwara  Norikaze,  of  1825  (Fig.  18).  The  early 
pictures  of  the  use  of  the  instrument  in  mercantile  affairs  are 
also  curious,  as  in  Fig.  19,  taken  from  an  anonymous  work  of 
the  seventeenth  century.  An  illustration  of  a  pupil  learning 
the  use  of  the  soroban,  from  Miyake  Kenryu's  work  of  1795* 
is  shown  in  Fig.  20. 

1  The  first  edition  was  1716. 


CHAPTER  IV. 
The  Sangi  applied  to  Algebra. 

As  stated  in  the  preceding  chapter,  it  seems  necessary  to 
break  the  continuity  of  the  historical  narative  by  speaking  of 
the  introduction  of  the  soroban  and  the  sangi,  since  these 
mechanical  devices  must  be  known,  at  least  in  a  general  way, 
before  the  contributions  of  the  later  writers  can  be  understood. 
As  already  explained,  the  chiknsaku  or  "bamboo  rods"  had 
been  brought  over  from  China  at  any  rate  as  early  as  600  A.  D., 
and  for  a  thousand  years  had  held  sway  in  the  domain  of 
calculation.  They  had  formed  one  of  the  inheritances  of  the 
people,  and  the  fact  that  they  are  still  used  in  Korea  shows 
how  strong  their  hold  would  naturally  have  been  with  a  patriotic 
race  like  the  Japanese.  We  have  much  the  same  experience 
in  the  Western  World  in  connection  with  the  metric  system 
today.  No  one  doubts  for  a  moment  that  this  system  will  in 
due  time  be  commonly  used  in  England  and  America,  the  race 
for  world  commerce  deciding  the  issue  even  if  the  merits  of 
the  system  should  fail  to  do  so.  Nevertheless  such  a  change 
comes  only  by  degrees  in  democratic  lands,  and  while  our 
complicated  system  of  compound  numbers  is  rapidly  giving 
way,  the  metric  system  is  not  so  rapidly  replacing  it. 

So  it  was  in  Japan  in  the  i/th  century.  The  samurai  despised 
the  plebeian  soroban,  and  the  guild  of  learning  sympathized  with 
this  attitude  of  mind.  The  result  was  that  while  the  soroban 
replaced  the  rods  for  business  purposes,  the  latter  maintained 
their  supremacy  in  the  calculations  of  higher  mathematics. 

There  was  a  further  reason  for  this  attitude  of  mind  in  the  fact 
that  the  rods  were  already  in  use  in  the  solution  of  the  equation, 


48  IV.  The  Sangi  applied  to  Algebra. 

having  been  well  known  for  this  purpose  ever  since  Ch'in  Chiu- 
shao(i247),  Li  Yeh  (1248  and  1257),  and  Chu  Chi-chieh  (1299)** 
had  described  them  in  their  works. 

As  stated  in  Chapter  III,  the  early  bamboo  rods  tended  to 
roll  off  the  table  or  out  of  the  group  in  which  they  were 
placed.  On  this  account  the  Koreans  use  a  trian'guloid  prism 
as  shown  in  the  illustration  on  page  22,  and  the  Japanese  in 
due  time  resorted  to  square  prisms  about  7  mm.  thick  and 
5  cm.  long.  These  pieces  had  the  name  sanc/m,  or,  more 
commonly,  sangi,  and  part  of  each  set  was  colored  red  and 
part  black,  the  former  representing  positive  mumbers  and  the 
latter  negative.  A  set  of  these  pieces,  now  a  rarity  even  in 
Japan,  is  shown  on  page  23. 

This  distinction  between  positive  and  negative  is  very  old. 
In  Chinese,  cheng  was  the  positive  and  fu  the  negative,  and 
the  same  ideographs  are  employed  in  Japan  today,  only  one  of 
the  terms  having  changed,  sei  being  used  for  cheng.  These 
Chinese  terms  are  found  in  the  Chiu-chang  Suan-shu  as  revised 
by  Chang  T'sang  in  the  2nd  century  B.  C, 2  and  hence  are 
probably  much  more  ancient  even  than  the  latter  date.  The 
use  of  the  red  and  black  for  positive  and  negative  is  found  in 
Liu  Hui's  commentary  on  the  Chiu-chang,  written  in  263  A.  D.,3 
but  there  is  no  reason  for  believing  that  it  originated  with  him. 
It  is  probably  one  of  the  early  mathematical  inheritances  of 
the  Chinese  the  origin  of  which  will  never  be  known.  As 
applied  to  the  solution  of  the  equation,  however,  we  have  no 
description  of  their  use  before  the  work  of  Ch'in  Chiu-shao  in 
1247.  In  the  treatises  of  Li  Yeh  and  Chu  Chi-chieh4  there  is 
given  a  method  known  as  the  fien-yuen-shu,  or  tengen  jutsu 

1  Chu  Shi-chieh,    or  Choo  Shi-ki.     Takebe's    commentary  (1690)  upon  his 
work   of  1299  is  mentioned  in  Chapter  VII.     He  also  wrote  in  1303  a  work 
entitled  Sze-yuen  yuh-kien,    "Precious   mirror   of  the  four  elements,"  but  this  is 
not  known  to  have  reached  Japan. 

2  See  No.  8  of  the  list  described  in  Chap.  II,  p.  II. 

3  See  p.  ii. 

4  His   work   was   known   as   Suan-hsiao   Chi-meng,   or  Sivan-hsiich-chi-mong. 
It  was    lost   to   the  Chinese   for  a  long  time,   but  Lo  Shih-lin  discovered  a 
Korean  edition  of  1660  and  reprinted  it  in  1839. 


IV.   The  Sangi  applied  to  Algebra.  49 

as  it  has  come  into  the  Japanese,  a  term  meaning  "The  method 
of  the  celestial  element." 

These  three  writers  appeared  in  widely  separated  parts  of 
China,  under  the  contending  monarchies  of  Song  and  Yuan, 
at  practically  the  same  time,  in  the  I3th  century.1  The  first, 
Ch'in  Chiu-shao,2  introduced  the  Monad  as  the  symbol  for  the 
unknown  quantity,  and  solved  certain  equations  of  the  6th, 
7th,  8th,  and  even  higher  degrees.  The  ancient  favorite  of 
the  West,  the  problem  of  the  couriers,  is  among  his  exercises. 
He  states  that  he  was  from  a  province  at  that  time  held  by 
the  Yuan  people  (the  Mongols). 

The  second  of  this  trio,  Li  Yeh,  3  wrote  "The  mirror  of  the 
mensuration  of  circles"  in  which  algebra  is  applied  to  trigono- 
metry.* The  third  of  the  group  is  Chu  Chi-chieh,  to  whose 
work  we  have  just  referred.  That  other  writers  of  prominence 
had  treated  of  algebra  before  this  time  is  evident  from  a  pas- 
sage in  the  preface  of  Chu  Chi-chieh's  work.  In  this  he  refers 
to  Chiang  Chou  Li  Wend,  Shih  Hsing-Dao,  and  Liu  Ju-Hsieh 
as  having  written  on  equations  with  one  unknown  quantity;  to 
Li  Te  Tsi,  who  used  equations  with  two  unknowns,  and  to  Liu 
Ta  Chien,  who  used  three  unknowns.  Chu  Chi-chieh 5  seems  to 
have  been  the  first  Chinese  writer  to  treat  of  systems  of  linear 
equations  with  four  unknowns,  after  the  old  "Nine  Sections." 


1  WYLIE,  A.,  Chinese  Researches,  Shanghai,  1897,  Part  III,  p.  175;  MIKAMI,  Y., 
A  Remark    on    the  Chinese  Mathematics    in  Cantor's  Geschichte    der  Mathematikt 
Archiv  der  Math,  und  Physik,  vol.  XV  (3),  Heft  I. 

2  Tsin    Kiu-tschau,    Tsin    Kew    Chaou.     His    work,    entitled   Su-shu   Chiu- 
chang,   or  Shu  hsiieh  Chi^t  Chang,   appeared  in  1247.    He  also  wrote  the  Shu 
shu  ta  Lueh  (General  rules  on  arithmetic). 

3  Or  Li-yay.     Li    was    the    family   name,    and   Yeh   or   Yay   the    personal 
name,    this    being    the    common    order.     He    is    also  known  by  his  familiar 
name,  Jin-king,  and  also  as  Li  Ching  Chai. 

4  His  two  works  are  entitled  T'se-yitan  Hai-ching  (1248)  and  I-ku  Yen-tuan 
(1257).     The  dates  are  a  little  uncertain,   since  Li  Yeh  states  in  the  preface 
that    the    second  work  was  printed  II  years  after  the  first.     Tse-yiian  means 
"to  measure  the  circle'',  and  Hai-ching  means  "mirror  of  sea". 

5  For  a  translation  of  his  work  I  am  indebted  to  Professor  Chen  of  Peking 
University.     D.  E.  S. 

4 


5O  IV.    The  Sangi  applied  to  Algebra. 

In  order  that  we  may  have  a  better  understanding  of  the 
basis  upon  which  Japanese  algebra  was  built,  a  few  words  are 
necessary  upon  the  state  to  which  the  Chinese  had  brought  the 
science  by  this  period.  While  algebra  had  been  known  before 
the  1  3th  century,  it  took  a  great  step  forward  through  the 
labors  of  the  three  men  whose  names  have  been  mentioned. 
They  called  their  method  by  various  names,  but  the  one  al- 
ready given,  and  Lih-tien-yiien-yih,  "The  setting  up  of  the  Ce- 
lestial Monad",  are  the  ones  commonly  used. 

In  general  in  this  new  algebra,  unity  represents  the  unknown 
quantity,  and  the  successive  powers  are  indicated  by  the  place, 
the  sangi  being  used  for  the  coefficients,  thus: 

= 

!==_     +  isx* 

TX  7T  +66x 


Li  Yeh  puts  the  absolute  term  on  the  bottom  line  as  here 
shown,  in  his  work  of  1248.  In  his  work  of  1259  and  in  the 
works  of  Ch'in  and  Chu  it  is  placed  at  the  top.  The  symbol 
after  66  was  called  yilen  and  indicated  the  monad,  while  the 
one  after  360  was  called  tai,  a  shortened  form  of  tai-kieJi,  "the 
extreme  limit".  In  practice  they  were  commonly  omitted.  The 
circle  is  the  zero  in  360,  and  the  cancellation  mark  indicates 
that  the  number  is  negative,  a  feature  introduced  by  Li  Yeh. 
With  the  sangi,  red  rods  would  be  used  for  i,  15,  and  66,  and 
black  ones  for  360.  It  will  be  noticed  that  this  symbolism  is 
in  advance  of  anything  that  was  being  used  in  Europe  at  this 
time,  and  that  it  has  some  slight  resemblance  to  that  used  by 
Bhaskara,  in  India,  in  the  I2th  century. 

Ch'in  Chiu-shao  (1247)  gives  a  method  of  approximating  the 
roots  of  numerical  higher  equations  which  he  speaks  of  as  the 
Ling-hmg-kae-fang,  "Harmoniously  alternating  evolution",  a  plan 
in  which,  by  the  manipulation  of  the  sangi,  he  finds  the  root 


IV.  The  Sangi  applied  to  Algebra.  51 

by  what  is  substantially  the  method  rediscovered  by  Homer, 
in  England,  in  1819.  Another  writer  of  the  same  period, 
Yang  Hwuy,  in  his  analysis  of  the  Chiu-chang*  gives  the 
same  rule  under  the  name  of  Tsang-ching-fang,  "Accumu- 
lating involution",  but  he  does  not  illustrate  it  by  solved 
problems.  We  are  therefore  compelled  to  admit  that  Horner's 
method  is  a  Chinese  product  of  the  I3th  century,  and  we 
shall  see  that  the  Japanese  adopted  it  in  what  we  have  called 
the  third  period  of  their  mathematical  history. 

It  is  also  interesting  to  know  that  Chu  Chi-chieh  in  the  Sze- 
yiien  Yu-kien  (1303)  gives  as  an  "ancient  method"  the  relation 
of  the  binomial  coefficients  known  in  Europe  as  the  "Pascal 
triangle",2  and  that  among  his  names  for  the  various  monads 
(unknowns)  is  the  equivalent  for  thing.*  This  is  the  same  as 
the  Latin  res  and  the  Italian  cosa,  both  of  which  had  un- 
doubtedly come  from  the  East.  It  is  one  of  the  many  interest- 
ing  problems  in  the  history  of  mathematics  to  trace  the  origin  v 
of  this  term.  * 

Chu  Chi-chieh  writes  the  equivalent  of  a  +  b  +  c  I 

+  x  as  is  here  shown,  except  that  we  use  T  for  i  T  I 
the  symbol  tai,  and  the  modern  numerals  instead  I 

of  the  sangi  forms.     The  square   of  this  expression  he  writes 

thus: 

I 

2      O       2 

2 

i     o    T    o     i 

2 

2       O       2 
I 

a  method  that  is  quickly  learned  and  easily  employed. 

*  See  p.  ir. 

2  This  was  also  known  in  Europe  long  before  Pascal.    See  SMITH,  D.  E., 
Kara  Arithmetica,  Boston,    1909,  p.  156. 

3  He    uses   the  names  heaven,    earth,    man,  thing,  although  the  first  three 
usually  designated  known  quantities. 

4  The    resemblance    to    the   Egyptian    ahe,    mass    (or  hait,    heap),    of  the 
Ahmes  papyrus,  c.  1700  B.  C,  will  possibly  occur  to  the  reader. 

4* 


52  IV.  The  Sangi  applied  to  Algebra. 

The  "celestial  element"  process  as  given  by  Chu  Chi-chieh 
in  1299  found  its  way  into  Japan  at  least  as  early  as  the 
middle  of  the  i/th  century,  and  the  Suan-Jisiao  Chi-meng  was 
reprinted  there  no  less  than  three  times.1  The  single  rule  laid 
down  in  this  classical  work  for  the  use  of  the  sangi  in  the 
solution  of  numerical  equations  contains  but  little  positive  infor- 
mation. Retaining  the  Japanese  terms,  and  translating  quite 
literally,  we  may  state  it  as  follows: — 

"Arrange  the  seki  in  the  jitsu  class,  adjusting  the  ho,  ren, 
and  gu  classes.  Then  add  the  like-signed  and  subtract  the 
unlike-signed,  and  evolve  the  root." 

This  reminds  one  of  the  cryptic  rules  of  the  Middle  Ages 
and  early  Renaissance  in  Europe,  but  unlike  some  of  these  it 
is  at  least  not  an  anagram  to  which  there  is  no  key.  The 
seki  is  the  quantity  in  a  problem  that  must  be  expressed  in 
the  absolute  term  before  solving,  and  which  is  represented  by 
the  sangi  in  next  to  the  top  row,  the  jitsu  class.  The  coeffi- 
cients of  the  first,  second,  and  third  powers  of  the  unknown 
are  then  represented  by  the  sangi  in  the  successive  rows  below, 
in  the  ho,  ren,  and  gu  classes.  The  rest  of  the  rule  amounts 
to  saying  that  the  pupil  should  proceed  as  he  has  been  taught. 
The  method  is  best  understood  by  actually  solving  a  numerical 
higher  equation,  but  inasmuch  as  the  manipulation  of  the  sangi 
has  already  been  explained  in  the  preceding  chapter,  the  coeffi- 
cients will  now  be  represented  by  modern  numerals.  The 
problem  which  we  shall  use  is  taken  from  the  eighth  book  of 
the  Tengen  Shinan  of  Sato  Moshun  or  Shigeharu,  published  in 
1698,  and  only  the  general  directions  will  be  given,  as  was  the 
custom.  The  reader  may  compare  the  work  with  the  common 
Horner  method  in  which  the  reasoning  involved  is  more  clear. 

Let  it  be  required  to  solve  the  equation 

II  520  —  432*—  236^  +  4*3  +  #4  =  0 


1  For  the  first  time  in  1658.  Dowun,  a  Buddhist  priest,  with  the  possible 
nom  de  plume  of  Baisho,  mentions  one  Hisada  (or  Kuda)  Gentetsu  (probably 
also  a  priest)  as  the  editor.  It  was  also  printed  in  1672  by  Hoshino  Jitsusen, 
and  some  time  later  by  Takebe  KenkO. 


IV.  The  Sangi  applied  to  Algebra.  53 

Arrange  the  sangi  on  the  board  to  indicate  the  following: 


(r) 
(o) 
(O 

(2) 

(3) 
(4) 

i 

i 

5 

2 

0 

— 

4 

3 

2 

— 

2 

3 

6 

4 

i 

Here  the  top  line,  marked  (r),  is  reserved  for  the  root,  and 
the  lines  marked  (o),  (i),  (2),  (3),  (4)  are  filled  with  the  sangi 
representing  the  coefficients  of  the  oth,  1st,  2d,  3d,  and  4th 
powers  of  the  unknown  quantity.  With  the  sangi,  the  negative 
432  and  236  would  be  in  black,  while  the  positive  1 1  520,  4, 
and  i  would  be  in  red. 

First  advance  the  ist,  2d,  3d,  and  4th  degree  classes  i,  2, 
3,  4  places  respectively,  thus: 


(r) 
(o) 
(O 
(2) 

(3) 
(4) 


I 

I 

5 

2 

O 

— 

4 

3 

2 

-2 

3 

6 

4 

I 

The  root  will  have  two  figures  and  the  tens'  figure  is  i. 
Multiply  this  10  by  the  i  of  class  (4)  and  add  it  to  class  (3), 
thus  making  14  in  class  (3).  Multiply  this  14  by  the  root,  10, 
and  add  it  to  — 236  of  class  (2),  thus  making  — 96  in  class  (2). 
Multiply  this  — 96  by  the  root,  10,  and  add  it  to  — 432  of  class 


54 


IV.  The  Sangi  applied  to  Algebra. 


(i),  thus  making  — 1392  in  class  (i).  Multiply  this  — 1392  by 
the  root,  10,  and  add  it  to  11520  of  class  (o),  thus  making 
— 2400.  The  result  then  appears  as  follows: 


(r) 
(o) 
(I) 
(2) 
(3) 
(4) 


i 

-2 

4 

o 

O 

-  I 

3 

9 

2 

-9 

6 

I 

4 

I 

Now  repeat  the  process,  multiplying  the  root,  10,  into  class  (4) 
and  adding  to  class  (3),  making  24;  multiply  24  by  the  root 
and  add  to  class  (2),  making  144;  multiply  144  by  the  root 
and  add  to  class  (i),  making  48.  The  result  then  appears  as 
follows: 


(r) 
(o) 
(i) 
(2) 
(3) 
(4) 


I 

-2 

4 

o 

0 

4 

8 

I 

4 

4 

2 

4 

I 

Repeat  the  process,  multiplying  the  root,  10,  into  class  (4) 
and  adding  to  class  (3),  making  34;  multiply  34  by  the  root 
and  add  to  class  (2)  making  484. 

Again  repeat  the  process,  multiplying  the  root  into  class  (4) 
and  adding  to  class  (3),  making  44. 

Now  move  the  sangi  representing  the  coefficients  of  classes 


IV.  The  Sangi  applied  to  Algebra. 


55 


(0,  (2X  (3)>  (4),  to  the  right  I,  2,  3,  4,  places,  respectively,  and 
we  have: 


(r) 
(o) 
(i) 
(2) 
(3) 
(4) 


i 

-2 

4 

0 

o 

4 

8 

4 

8 

4 

4 

4 

i 

The  second  figure  of  the  root  is  2. *  Multiply  this  into  class 
(4)  and  add  to  class  (3),  making  46.  Multiply  the  same  root 
figure,  2,  into  this  class  (3)  and  add  to  class  (2),  making  576. 
Multiply  this  576  by  2  and  add  to  class  (I),  making  1200. 
Multiply  this  1200  by  2  and  add  to  class  (o),  making  o.  The 
work  now  appears  as  follows: — 


(r) 
(o) 
(i) 

(2) 

(3) 

(4) 

i 

2 

i 

2 

5 

7 

6 

4 

6 

i 

The  root  therefore  is  12. 

It  may  now  be  helpful  to  give  a  synoptic  arrangement  of 
the  entire  process  in  order  that  this  Chinese  method  of  the 
1 3th  century,  practiced  in  Japan  in  the  I7th  century,  may  be 


1  It  is  not  stated  how  either  figure  is  ascertained. 


56  IV.  The  Sangi  applied  to  Algebra. 

compared  with  Horner's  method.      The  work   as  described  is 
substantially  as  follows: 

Given  x*  +  4**  —  2^6x2  —  432^  +  1 1  520  =  o 
i+    4  —  236--    432+11520 
10       140 —    960 — 13920 


I 

14- 

IO 

-96  — 
24O 

1392  — 
1440 

24OO 

I 

24 

IO 

144 

340 

48- 

24OO 
2400 

I 

34 

10 

484 

48 
1152 

0 

I 

44 

2 

484 
92 

I20O 

I  46     576 

Chu  Chi-chieh  also  gives,  in  the  Suan-hsiao  Chi-ineng^  rules 
for  the  treatment  of  negative  numbers.  The  following  transla- 
tions are  as  literal  as  the  circumstances  allow: 

"When  the  same-named  diminish  each  other,  the  different- 
named  should  be  added  together.1  If  then  there  is  no  opponent 
for  a  positive  term,  make  it  negative;  and  for  a  negative,  make 
it  positive." 2 

"When  the  different-named  diminish  each  other  the  same- 
named  should  be  added  together.  If  then  there  is  no  opponent 
for  a  positive,  make  it  positive;  and  for  a  negative,  make  it 
negative."  3 

"When  the  same-named  are  multiplied  together,  the  product 
is  made  positive.  When  the  different-named  are  multiplied 
together,  the  product  is  made  negative." 

The  method  of  the  "celestial  element",  with  the  sangi,  and 
with  the  rules  just  stated,  entered  into  the  Japanese  mathe- 


1  This    is    intended   to  mean  that  when  (+ 4)  —  (+ 3)=  +  (4  —  3\    then 
(+  4)  -  (—  3)  should  be  +  4  +•  3- 

2  That  is,  o  —  (-f-  4)  =  —  4,  and  o  —  (—  4)  =  -f-  4. 

3  When  (+  p)  -  (—  q)  =  +  p  +  q,  then  (—  p)  -  (-f-  q)  =  -  (p  +  q>    Also, 

=  +  4»  and  o+(— 4)=— 4. 


IV.  The  Sangi  applied  to  Algebra. 


57 


matics  of  the  iyth  century,  to  be  described  in  the  following 
chapter.  They  were  purely  Chinese  in  origin,  but  Japan  ad- 
vanced the  method,  carrying  it  to  a  high  degree  of  perfection 
at  the  time  when  China  was  abandoning  her  native  mathe- 
matics under  the  influence  of  the  Jesuits.  It  is,  therefore,  in 
Japan  rather  than  China  that  we  must  look  in  the  iyth  cen- 
tury for  the  strictly  oriental  development  of  calculation,  of  al- 
gebra, and  of  geometry. 

Among  the  other  writers  of  the  period  several  treated  of 
magic  squares.  Among  these  was  Hoshino  Sanenobu,  whose 
Ko-ko'gen  Sho  (Triangular  Extract)  appeared  in  1673.  Half  of 
one  of  his  magic  squares  in  shown  in  the  following  facsimile : 


-t- 


•f 


A 


W 


It 


if 


f 


S 


w 


1" 


1L 


•ft 


-ft 


it 


. 


I 


It 


•f 


-f 


ea 


A 


71- 


I    A 


¥    5 


Fig.  21.     Half  of  a  magic   square,   from  Hoshino  Sanenobu's   work   of  1673. 


One  who  is  not  of  the  Japanese  race  cannot  refrain  from  mar- 
velling at  the  ingenuity  of  many  of  these  problems  proposed 
during  the  i/th  century,  and  at  the  painstaking  efforts  put 
forth  in  their  solution.  He  is  reminded  of  the  intricate  ivory 


58  IV.  The  Sangi  applied  to  Algebra. 

carvings  of  these  ingenious  and  patient  people,  of  the  curious 
puzzles  with  which  they  delight  the  world,  and  of  the  finish 
which  characterizes  their  artistic  productions.  Few  of  these 
problems  could  be  mistaken  for  western  productions,  and  the 
solutions,  so  far  as  they  are  given,  are  like  the  art  and  the 
literature  of  the  people,  indigenous  to  the  soil  of  Japan. 


CHAPTER  V. 
The  Third  Period. 

It  was  stated  in  the  opening  chapter  that  the  third  of  the 
periods  into  which  we  arbitrarily  divide  the  history  of  Japanese 
mathematics  was  less  than  a  century  in  duration,  extending 
from  about  1600  to  about  1675.  The  first  of  these  dates  is 
selected  as  marking  approximately  the  beginning  of  the  activity 
of  Mori  Kambei  Shigeyoshi,  who  was  mentioned  in  Chapter  III, 
and  the  last  as  marking  that  of  Seki.  It  was  an  era  of 
intellectual  awakening  in  Japan,  of  the  welcoming  of  Chinese 
ideas,  and  of  the  encouragement  of  native  effort.  Of  the  work 
of  Mori  we  have  already  spoken,  because  he  had  so  much  to 
do  with  making  known,  and  possibly  improving,  the  soroban. 
It  now  remains  to  speak  of  his  pupils,  and  first  of  Yoshida. 

Yoshida  Shichibei  Koyu,  or  Mitsuyoshi,  was  born  at  Saga, 
near  Kyoto,  in  1 598,  as  we  are  told  in  Kawakita's  manuscript, 
the  Honchd  Siigakii  Sliiryo.  He  belonged  to  an  ancient  family 
that  had  contributed  not  a  few  illustrious  names  to  the  history 
of  the  country.  Yoshida  Sokei,  for  example,  who  died  in  1572, 
was  well  known  in  medicine,  and  had  twice  made  a  journey  to 
China  in  search  of  information,  once  with  a  Buddhist  bonze1 
in  1539,  and  again  in  1547.  His  son  Koko,  (1554 — 1616),  was 
a  noted  engineer,  and  is  known  for  his  work  in  improving 
navigation  on  the  Fujikawa  and  other  rivers  that  had  been 
too  dangerous  for  the  passage  of  boats.  Koko's  son  Soan 
was,  like  his  father,  well  known  for  his  learning  and  for  his 
engineering  skill. 2  Yoshida  Koyu,  the  mathematician,  was  a 

*  Priest.     The  name  is  a  Portuguese  corruption  of  a  Japanese  term. 
2  See  the  Sentetsu  Sodan  Zoku-hen,  1884,  Book  I. 


6O  V.  The  Third  Period. 

grandson,  on  his  mother's  side,  of  Yoshida  Koko.1  He  was 
also  related  in  another  way  to  the  Yoshida  family,  being  the 
eldest  son  of  Yoshida  Shuan,  who  was  the  great-grandson  of 
Sokei's  father,  Sochu. 

Yoshida,  as  we  shall  now  call  him,  early  manifested  a  taste 
for  mathematics,  going  as  a  youth  to  Kyoto  that  he  might 
study  under  the  renowned  Mori.  His  ignorance  of  Chinese  was 
a  serious  handicap,  however,  and  his  progress  was  a  disap- 
pointment. He  thereupon  set  to  work  to  learn  the  language, 
studying  under  the  guidance  of  his  relative  Yoshida  Soan,  and 
in  due  time  became  so  proficient  that  he  was  able  to  read  the 
Suan-fa  Tung-tsong  of  Ch'eng  Tai-wei. 2  His  progress  in 
mathematics  then  became  so  rapid  that  it  is  related  3  that  he 
soon  distanced  his  master,  so  that  Mori  himself  was  glad  to 
become  his  pupil.  Yoshida  also  continued  to  excel  in  Chinese, 
so  that,  whereas  Mori  knew  the  language  only  indifferently, 
his  quondam  pupil  became  master  of  the  entire  mathematical 
literature. 

Mori's  works  were  the  earliest  native  Japanese  books  on 
mathematics  of  which  we  have  any  record,  but  they  seem  to 
be  irretrievably  lost.  It  is  therefore  to  Yoshida  that  we  look 
as  the  author  of  the  oldest  Japanese  work  on  mathematics 
extant.  This  work  was  written  in  1627  and  is  entitled  Jinko- 
ki.  The  name  is  interesting,  the  Chinese  ideogram  jin  meaning 
(among  other  things)  a  small  number,  ko  meaning  a  large 
number,  and  ki  a  treatise,  so  that  the  title  signifies  a  treatise 
on  numbers  from  the  greatest  to  the  least.  Yoshida  tells  us 
in  the  preface  that  it  was  selected  for  him  by  one  Genko,  a 
Buddhist  priest,  and  it  is  typical  of  the  condensed  expressions 
of  the  Japanese. 

The  work  relates  chiefly  to  the  arithmetical  operations  as 
performed  on  the  soroban,  including  square  and  cube  root,  but 
it  also  has  some  interesting  applications  and  it  gives  3.16  for 


1  ENDO,  Book  I,  p.  35. 

2  Which  had  appeared  in  1593.     See  p.  34. 

3  By  KAWAKITA  in  the  Honcho  Sugaku  Shiryo. 


Vr.  The  Third  Period.  6 1 

the  value  of  TT.  It  is  based  largely  upon  the  Suan-fa  T'ung- 
tsong  already  described,  and  the  preface  states  that  it  originally 
consisted  of  eighteen  books.  Only  three  books  have  come 
down  to  us,  however,  and  indeed  we  are  assured  that  only 
three  were  ever  printed. :  This  was  the  first  treatise  on  mathe- 
matics ever  printed  in  Japan,  or  at  least  the  first  of  any  im- 
portance. *  It  appeared  in  16273  and  was  immediately  received 
with  great  enthusiasm.  Even  during  Yoshida's  life  a  number 
of  editions  appeared,4  and  the  name  Jinko-ki  was  used  so 
often  after  his  death,  by  other  authors,  that  it  became  a  syno- 
nym for  arithmetic,  as  algorismus  did  in  Europe  in  the  late 
Middle  Ages.s  Indeed  it  is  hardly  too  much  to  compare  the 
celebrity  of  the  Jinko-ki  in  Japan  with  that  of  the  arithmetic 
of  Nicomachus  in  the  late  Greek  civilization.  Yoshida  also 
wrote  on  the  calendar,  but  these  works6  were  not  so  well 
known. 

So  great  was  the  fame  of  Yoshida  that  he  was  called  to 
the  court  of  Hosokawa,  the  feudal  lord  of  Higo,  that  he  might 
instruct  his  patron  in  the  art  of  numbers  Here  he  resided  for 
a  time,  and  at  his  lord's  death,  in  1641,  he  returned  to  his 
native  place  and  gathered  about  him  a  large  number  of  pupils, 
even  as  Mori  had  done  before  him.  In  his  declining  years  an 
affection  of  the  eyes,  which  had  troubled  him  from  his  youth, 
became  more  serious,  and  finally  resulted  in  the  affliction  of 


1  By  the   bonze  GenkO   who   wrote   the   preface,   and   by  Yoshida  himself 
at  the  end  of  the  1634  edition. 

2  Mr.  ENDO  has   shown   the    authors   the   copy   of  the    edition  of  1634  in 
the    library    of   the    Tokyo"    Academy    and   has    assured   us    that   the    edition 
of  1627  was  the  first  Japanese  mathematical  work  of  any  importance.    There 
is  a  tradition  that  MORI'S  Kijo  Ranjo  was  also  printed. 

3  That  is,  the  4th  year  of  Kwan-ei. 

4  As  in  1634,  1641,  and   1669,  all  edited  by  Yoshida.     There  were  several 
pirated  editions.    See  MURAMATSU'S  Sanso  of  1663,  Book  III;  ENDO,  Book  I, 
PP-  58,  59,  84  etc. 

5  Compare  the  German  expression  "Nach  Adam  Riese",  the  English  "Accord- 
ing to  Cocker",  the  early  American  "According  to  Daboll",  and  the  French 
word  Bareme. 

6  For  example,  the  IVakan  Go-un  and  the  Koreki  Benran. 


62  V.  The  Third  Period. 

total  blindness, — the  fate  of  Saunderson  and  of  Euler  as  well. 
He  died  in  1672  at  the  age  of  seventy-four.1 

The  immediate  effect  of  the  work  of  Mori  and  Yoshida  was 
a  great  awakening  of  interest  in  computation  and  mensuration. 
In  1630  the  Shogun  established  the  Kobun-in,  a  public  school 
of  arts  and  sciences.  Unfortunately,  however,  mathematics 
found  no  place  in  the  curriculum,  remaining  in  the  hands  of 
private  teachers,  as  in  the  days  of  the  German  Rechenmeister. 
Nevertheless  the  science  progressed  in  a  vigorous  manner  and 
numerous  books  were  published  upon  the  subject.  Yoshida 
had  appended  to  one  of  the  later  editions  of  his  Jinko-ki  a 
number  of  problems  with  the  proposal  that  his  successors 
solve  them.  These  provoked  a  great  deal  of  discussion  and 
interest,  and  led  other  writers  to  follow  the  same  plan,  thus 
leading  to  the  so-called  idai  skoto,2  "mathematical  problems 
proposed  for  solution  and  solved  in  subsequent  works".  This 
scheme  was  so  popular  that  it  continued  until  1813,  appearing 
for  the  last  time  in  the  Sangaku  Kochi  of  Ishiguro  Shin-yu. 

The  particular  edition  of  Yoshida's  Jinko-ki  in  which  these 
problems  appeared  is  not  extant,  but  the  problems  are  known 
through  their  treatment  by  later  writers,  and  some  of  them 
will  be  given  when  we  come  to  speak  of  the  work  of  Isomura. 

The  second  of  Mori's  "three  honorable  scholars"  mentioned 
in  Chapter  III  was  Imamura  Chisho,  and  twelve  years  after  the 
appearance  of  the  Jinko-ki,  that  is  in  1639,  ne  published  a 
treatise  entitled,  Jugai-roku.*  Yoshida's  work  had  appeared 
in  Japanese,  although  it  followed  the  Chinese  style,  but  Ima- 
mura wrote  in  classical  Chinese.  Beginning  with  a  treatment 
of  the  soroban,  he  does  not  confine  himself  to  arithmetic,  as 
Yoshida  had  done,  but  proceeds  to  apply  his  number  work  to 
the  calculations  of  areas  and  volumes,  as  in  the  case  of  the 

1  C.  KAWAKITA,  Honcho  Sugaku  Shiryo;  ENDO,  Book  I,  p.  84. 

2  A  term  used  by  later  scholars. 

3  Mr.  Endo  has  shown  the   authors  a   copy  of  Ando's  commentary  in  the 
library  of  the  Academy  of  Science    at  Tokyo,  and  Dr.  K.  Kano   has  a  copy 
of  the  original  at  present  in  his  valuable  library.     At  the    end    of   the  work 
the  author  states  that  only  a  hundred  copies  were  printed. 


V.  The  Third  Period.  63 

circle,  the  sphere,  and  the  cone.  While  Yoshida  had  taken 
3. 1 6  for  the  value  of  TT,  Imamura  takes  3. 162.  Ando  Yuyeki 
of  Kyoto  refers  to  this  in  his  Jugai-roku  Kana-sho,  printed  in 
1660,  as  obtained  by  extracting  the  square  root  of  10.  If  this 
is  true,  Yoshida  obtained  his  in  the  same  way,  the  square  root 
of  10  having  long  been  a  common  value  for  TT  in  India  and 
Arabia,  as  well  as  in  China.  Liu  Hui's  commentary  on  the 
"Nine  Sections"  asserts  that  the  first  Chinese  author  to  use 
this  value  was  Chang  Heng,  78 — 139  A.  D.  It  was  also 
used  by  Ch'en  Huo  in  the  eleventh  century,  and  by  Ch'in 
Chiu-shao  in  his  Su-shu  Chiu-chang  of  1247.'  Some  Chinese 
writers  even  in  the  present  dynasty  have  used  it,  and  it  was 
very  likely  brought  from  that  country  to  Japan.  It  is  of  interest 
to  note  that  lumbermen  and  carpenters  in  certain  parts  of 
Japan  use  this  value  at  the  present  time. 

Imamura  gives  as  a  rule  for  finding  the  area  of  a  circle 
that  the  product  of  the  circumference  by  the  diameter  should 
be  divided  by  4.  The  volume  of  the  sphere  with  diameter 
unity  is  given  as  0.51,  which  does  not  fit  his  value  of  rr  as 
closely  as  might  have  been  expected.  He  also  gives  a  number 
of  problems  about  the  lengths  of  chords,  and  writes  extensively 
upon  \hQKaku-jutsu  or  "polygonal  theory", — calculations  relating 
to  the  regular  polygons  from  the  triangle  to  the  decagon.  This 
theory  attracted  considerable  attention  on  the  part  of  his  suc- 
cessors and  added  much  to  Imamura's  reputation.2  This 
treatise  was  translated  into  Japanese  and  a  commentary  was 
added  by  Imamura's  pupil,  Ando  Yuyeki,  in  1660. 

The  year  following  the  appearance  of  the  original  edition 
Imamura  published  the  Inki  Sanka  (1640),  a  little  work  on  the 
soroban,  written  in  verse.  The  idea  was  that  in  this  way  the 
rules  could  the  more  easily  be  memorized,  an  idea  as  old  as 
civilization.  The  Hindus  had  followed  the  same  plan  many 


1  MIKAMI,  Y.,    On  the  development  of  the  Chinese  mathematics   (in  Japanese), 
in  the  Journal  of  the  Tokyo  Physics  School,  No.  203,  p.  450;  Mathematical  papers 

from  the  Far  East,  Leipzig,  19 to,  p.  5. 

2  ENDO,  Book  I,  pp.  59,  60. 


64 


V.  The  Third  Period. 


centuries  earlier,  and  a  generation  or  so  before  Imamura  wrote 
it  was  being  followed  by  the  arithmetic  writers  of  England. 

The  third  of  the  San-ski  of  Mori  was  Takahara  Kisshu,  also 
known  as  Yoshitane.1  While  he  contributed  nothing  in  the 
way  of  a  published  work,  he  was  a  great  teacher  and  numbered 
among  his  pupils  some  of  the  best  mathematicians  of  his  time. 

During  this  period  of  activity  numerous  writers  of  prominence 
appeared,  particularly  on  the  soroban  and  on  mensuration. 
Among  these  writers  a  few  deserve  a  brief  mention  at  this 
time.  Tawara  Kamei  wrote  his  Shinkan  Sampo-ki  in  1652, 


wii 

£*H? 


O 


v 


,..      Fig.  22.     From  Yamada's  Kaisan-ki  (1656),  showing  a  rude  trigonometry. 

and  Yenami  Washo  his  Sanryo-roku  in  the  following  year.  In 
/vC-i656  Yamada  Jusei  published  the  Kaisan-ki  (Fig.  22)  which 
was  very  widely  read,  and  the  title  of  which  was  adopted, 
with  various  prefixes,  by  several  later  writers.  The  following 
year  (1657)  saw  the  publication  of  Hatsusaka's  Yempo  Shikan- 
ki  and  Shibamura's  Kakuchi  Sansho.  A  year  later  (1658) 
appeared  Nakamura's  Shikaku  Mondo,  followed  in  1660  by 
\satauT2iS-Ketsugt-sk0,  in  1663  by  Muramatsu's  Sanso,  in  1664 


1  The  names  are  synonyms. 


V.  The  Third  Period.  65 

by  Nozavva  rlQ\c\\6'sJDdkal-shd,  and  in  1666  by  Sato's  Kongenki. 
These  are  little  more  than  names  to  Western  readers,  and  yet 
they  go  to  show  the  activity  that  was  manifest  in  the  field  of 
elementary  mathematics,  largely  as  the  result  of  the  labors  of 
Mori  and  of  Yoshida.  The  works  themselves  were  by  no 
means  commercial  arithmetics,  for  they  perfected  little  by  little 
the  subject  of  mensuration,  the  method  of  approximating  the 
value  of  IT,  and  the  treatment  of  the  regular  polygons,  besides 
offering  a  considerable  insight  into  the  nature  of  magic  squares 
and  magic  circles.  To  these  books  we  are  indebted  for  our 
knowledge  of  the  work  of  this  period,  and  particularly  to  the 
Kaisan-ki  (1656),  the  Shikaku-Mondo  (1658),  and  the  Ketsugi- 
sho,  (1660). 

The  last  mentioned  work,  the  Ketsugi-sho-,  was  written  by 
a  pupil  of  Takahara  Kisshu,1  who  was  one  of  the  San-ski  oi 
Mori.  His  name  was  Isomura2  Kittoku,  and  he  was  a  native 
of  Nihommatsu  in  the  north-eastern  part  of  Japan.  Isomura's 
Ketsugi-sJw*  appeared  in  five  books  in  1660,  and  was  again 
published  in  1684  with  notes.  We  know  little  of  his  life,  but 
he  must  have  been  very  old  when  the  second  edition  of  his 
work  appeared  for  he  tells  us  in  the  preface  that  at  that  time 
he  could  hardly  hold  a  soroban  or  the  sangi. 

Two  features  of  the  Ketsugi-slio  deserve  mention, — Isomura's 
statement  of  the  Yoshida  problems  (including  an  approach  to 
integration,  as  seen  in  Fig.  23)  and  similar  ones  of  his  own, 
and  his  treatment  of  magic  squares  and  circles.  Each  of  these 
throws  a  flood  of  light  upon  the  nature  of  the  mathematics  of 
Japan  in  its  Renaissance  period,  just  preceding  the  advent  of 
the  greatest  of  her  mathematicians,  Seki,  and  each  is  therefore 


1  OZAWA,  Sanka  Furyaku,  "Brief  Lineage  of  Mathematicians",  manuscript 
of  1801. 

2  ENDO  gives  it  as  ISOMURA,  Book  I,  pp.  65,  67,   and   Book  II,  p.  20  etc., 
and  in   this   he   was   at   first   followed   by   HAYASHI,  History,  part  I,    p.  33, 
although  the  latter  soon  after  discovered  that  IWAMURA  was  the  better  form. 
HAYASHI  gives  the  personal  name  as  Yoshinori. 

3  Or  Sampo-kelsugi-sho. 

5 


66 


V.  The  Third  Period. 


worthy  of  our  attention.    Of  the  Yoshida  problems  the  following 
are  types:1 

"There  is  a  log  of  precious  wood  18  feet2  long,  whose  bases 
are  5  feet  and  2^  feet  in  circumference.  ...  Into  what  lengths 
should  it  be  cut  to  trisect  the  volume?" 

"There  have  been  excavated  560  measures  of  earth  which 
are  to  be  used  for  the  base  of  a  building.  3  The  base  is  to 
be  30  measures  square  and  9  measures  high.  Required  the 
size  of  the  upper  base." 


Fig.  23.     From  the  second  (1684)  edition  of  Isomura's  Ketsugi-sho. 

"There  is  a  mound  of  earth  in  the  form  of  the  frustum  of 
a  circular  cone.  The  circumferences  of  the  bases  are  40  mea- 
sures and  1 20  measures,  and  the  mound  is  6  measures  high. 
If  1 200  measures  of  earth  are  taken  evenly  off  the  top,  what 
will  then  be  the  height?" 

"A  circular  piece  of  land  100  measures  in  diameter  is  to  be 
divided  among  three  persons  so  that  they  shall  receive  2900, 


1  The  Ketsugi-sho  of  1660,  Book  4. 

2  In  the  original  "3  measures". 

3  That  is,  for  a  mound  in  the  form   of  a  frustum  of  a  square  pyramid. 


V.  The  Third  Period.  67 

2500,  and  2500  measures  respectively.1  Required  the  lengths 
of  the  chords  and  the  altitudes  of  the  segments." 

The  rest  of  the  problems  relate  to  the  triangle  and  to  linear 
simultaneous  equations  of  the  kind  found  in  such  works  as  the 
"Nine  Sections",  the  Suan-fa  Tung-tsong,  and  the  Suan-hsiao 
Chi-meng.  The  last  of  the  problems  given  above  is  solved  by 
Isomura  as  follows: — 

"Divide  7900  measures,2  the  total  area,  by  2900  measures 
of  the  northern  segment,  the  result  being  2 724. 3  Double  this 
result  and  we  have  5448.  Divide  the  square  of  the  diameter, 
100  measures,  by  5448  and  the  result  is  1835.554*  measures. 
The  square  root  of  this  is  42.85  measures.  Subtract  this  from 
half  the  diameter  and  we  have  7.15  measures.  Multiply  the 
42.85  by  this  and  we  have  306.4  measures.  We  now  multiply 
by  a  certain  constant  for  the  square  and  the  circle,  and  divide 
by  the  diameter  and  we  have  3.45  measures.  Subtract  this 
from  42.85  measures  and  we  have  39.4  measures  for  the 
height  of  the  northern  segment . . ." 

Following  Yoshida's  example,  Isomura  gives  a  series  of 
problems  for  solution,  a  hundred  in  number,  placing  them  in 
his  fifth  book.  A  few  of  these  will  show  the  status  of  mathe- 
matics at  the  time  of  Isomura: 

"From  a  point  in  a  triangle  lines  are  drawn  to  the  vertices. 
Given  the  lengths  of  these  lines  and  of  two  sides  of  the  triangle, 
to  find  the  length  of  the  third  side  of  the  triangle."  (No.  28.) 

"A  string  62.5  feet  long  is  laid  out  so  as  to  form  Seimei's 
Seal,  s  Required  the  length  of  the  side  of  the  regular  pentagon 
in  the  center."  (No.  38.) 

"A  string  is  coiled  so  as  first  to  form  a  circle  0.05  feet  in 
diameter,  and  [then  so  that  the  coils  shall]  always  keep  0.05 
feet  apart,  and  the  coil  finally  measures  125  feet  in  diameter. 


1  By  drawing  two  parallel  chords. 

2  It  would  have  been  7854  if  he  had  taken  ir=  3.1416. 

3  I.  e.,  2.724+- 

4  Where  we  now  introduce  the  fraction  for  clearness. 

5  Abe  no  Seimei   was   a   famous   astrologer  who   died   in   1005.     His  seal 
was  the  regular  pentagonal  star,  the  badge  of  the  Pythagorean  brotherhood. 

5* 


68  V.  The  Third  Period. 

What  is  the  length  of  the  string?"  (No.  39.)  The  reading 
of  this  problem  is  not  clear,  but  Isomura  seems  to  mean  that 
a  spiral  of  Archimedes  is  to  be  formed  coiled  about  an  inner 
circle,  and  finally  closing  in  an  outer  circle.  The  curve  has 
attracted  a  good  deal  of  attention  in  Japan. 

"There  is  a  log  18  feet  long,  the  diameter  of  the  extremities 
being  I  foot  and  2.6  feet  respectively.  This  is  wound  spirally 
with  a  string  75  feet  long,  the  coils  being  2.5  feet  apart.  How 
many  times  does  the  string  go  around  it?"  (No.  41.) 

"The  bases  of  a  frustum  of  a  circular  cone  have  for  their 
respective  diameters  50  measures  and  120  measures,  and  the 
height  of  the  frustum  is  1 1  measures.  Required  to  trisect  the 
volume  by  planes  perpendicular  to  the  base."  (No.  44.) 

"The  bases  of  a  frustum  of  a  circular  cone  have  for  their  re- 
spective diameters  120  and  250  measures,  and  the  height  of  the 
frustum  is  25  measures.  The  frustum  is  to  be  cut  obliquely. 
Required  the  perimeter  of  the  section."  (No.  45.)  Presumably 
the  cutting  plane  is  to  be  tangent  to  both  bases,  thus  forming 
a  complete  ellipse,  a  figure  frequently  seen  in  Japanese  works. 

"In  a  circle  3  feet  in  dia- 
meter 9  other  circles  are  to  be 
placed,  each  being  0.2  of  a 
foot  from  every  other  and  from 
the  large  circle.  Required  the 
diameter  of  the  larger  circle  in 
the  center,  and  of  the  smaller 
circles  surrounding  it."  (No.  60.) 
This  requires  us  to  place  a 
circle  A  in  the  center,  ar- 
ranging eight  smaller  circles  B 
about  it  so  as  to  satisfy  the 
conditions. 

"If  19  equal  circles  are  described  outside  a  given  circle  that 
has  a  circumference  of  12  feet,  so  as  to  be  tangent  to  the 
given  circle  and  to  each  other;  and  if  19  others  are  similarly 
described  within  the  given  circle,  what  will  be  the  diameters 
of  the  circles  in  these  two  groups?"  (No.  61.) 


V.  The  Third  Period. 


69 


"To  find  the  length  of  the  minor  axis  of  an  ellipse  whose 
area  is  748.940625,  and  whose  major  axis  is  38  measures." 
(No.  84.) 

"To  find  one  axis  of  an  ellipsoid  of  revolution,  the  other 
axis  being  1.8  feet,  and  the  volume  being  2422,  the  unit  of 
volume  being  a  cube  whose  edge  is  o.i  of  a  foot."  (N.  85.) 
Here  the  major  axis  is  supposed  to  be  the  axis  of  revolution. 

Isomura  was  also  interested  in  magic  squares,  and  these  forms 
were  evidently  the  object  of  much  study  in  his  later  years, 
since  the  1684  edition  of  his  Ketsugi-sho  contains  considerable 
material  relating  to  the  subject.  In  the  first  edition  (1660) 
there  appear  both  odd  and  even-celled  squares.  The  following 
types  suffice  to  illustrate  the  work.1 


40 

38 

2 

6 

i 

42 

46 

4i 

20 

17 

37 

19 

32 

9 

3 

16 

26 

21 

28 

34 

47 

39 

36 

27 

25 

23 

14 

1  1 

43 

35 

22 

29 

24 

15 

7 

5 

18 

33 

13 

3i 

30 

45 

4 

12 

48 

44 

49 

8 

10 

55 

4 

2 

62 

64 

60 

6 

7 

5i 

20 

22 

17 

50 

42 

44 

14 

9 

49 

40 

28 

25 

37 

16 

56 

12 

46 

29 

3i 

34 

36 

19 

53 

13 

18 

35 

33 

32 

30 

47 

52 

54 

41 

26 

38 

39 

27 

24 

ii 

8 

21 

43 

48 

15 

23 

45 

57 

58 

61 

63 

3 

i 

5 

59 

10 

1  It  should  be  said  that  the  history  of  the  magic  square  has  never  ade- 
quately been  treated.  Such  squares  seem  to  have  originated  in  China  and 
to  have  spread  throughout  the  Orient  in  early  times.  They  are  not  found 
in  the  classical  period  in  Europe,  but  were  not  uncommon  during  and  after 
the  1 2th  century.  They  are  used  as  amulets  in  certain  parts  of  the  world, 
and  have  always  been  looked  upon  as  having  a  cabalistic  meaning.  For  a 
study  of  the  subject  from  the  modern  standpoint  see  ANDREWS,  W.  S.,  Magic 
Squares,  Chicago,  1907,  and  subsequent  articles  in  The  Open  Court. 


V.  The  Third  Period. 


51 

46 

53 

6 

i 

8 

69 

64 

7i 

52 

50 

48 

7 

5 

3 

70 

68 

66 

47 

54 

49 

2 

9 

4 

65 

72 

67 

60 

55 

62 

42 

37 

44 

24 

'9 

26 

61 

59 

57 

43 

4i 

39 

25 

23 

21 

56 

63 

58 

38 

45 

40 

20 

27 

22 

IS 

10 

17 

78 

73 

80 

33 

28 

35 

16 

14 

12 

79 

77 

75 

34 

32 

30 

1  1 

18 

13 

74 

Si 

76 

29 

36 

3i 

92 

9i 

i5 

89 

4 

84 

14 

99 

I  I 

6 

13 

73 

22 

20 

80 

83 

78 

24 

25 

88 

85 

69 

38 

40 

35 

68 

60 

62 

32 

16 

3 

27 

67 

58 

46 

43 

55 

34 

74 

98 

96 

30 

64 

47 

49 

52 

54 

37 

71 

5 

8 

3i 

36 

53 

5i 

50 

48 

65 

70 

93 

18 

72 

59 

44 

56 

57 

45 

42 

29 

83 

94 

26 

39 

61 

66 

33 

4i 

63 

75 

7 

i 

76 

79 

81 

21 

19 

23 

77 

28 

IOO 

95 

10 

86 

12 

97 

17 

87 

2 

90 

9 

In  the  last  (1684)  edition  he  gives  a  number  of  new  arrange- 
ments, including  the  following: 


V.  The  Third  Period. 


5 

23 

16 

4 

25 

15 

14 

7 

18 

II 

24 

17 

13 

9 

2 

20 

8 

19 

12 

6 

I 

3 

10 

22 

21 

IO 

8 

35 

33 

24 

i 

19 

26 

17 

15 

6 

28 

5 

12 

30 

34 

16 

H 

23 

21 

3 

7 

25 

32 

18 

31 

22 

20 

ii 

9 

36 

13 

4 

2 

29 

27 

Isomura   did    also    a    good   deal  of  work  on  magic  circles, 
the  following  appearing  in  his   1660  edition: 


V.  The  Third  Period. 


V.  The  Third  Period. 


In  the  1684  edition 
of  his  Ketsugi-sho  he 
gives  what  he  calls 
sets  of  magic  wheels. 
Here,  and  on  pages 
74  and  75,  the  sums 
in  the  minor  circles 
are  constant. 

Isomura's  method1 
of  finding  the  area 
of  the  circle  is  as 

1  1660  edition  of  the 
Ketsugi-sho,  Book  III. 


74 


V.  The  Third  Period. 


follows:  Take  a  circle  of  diameter  10  units,  and  divide  the 
circumference  into  parts  whose  lengths  are  each  a  unit.  It 
will  then  be  found  that  there  are  31  of  these  equal  arcs,  with 
a  smaller  arc  of  length  0.62.  Join  the  points  of  division  to  the 
center,  thus  making  a  series  of  triangular  shaped  figures.  By 


(28 


(28 


do) 


(13) 


(37; 


(26 


(121 


38. 


wo; 


(39) 


119} 


(31, 


(21; 


(25) 


(16) 


(20) 


(29) 


27) 


(35; 


(1V) 


(30J 


(11J 


(331 


dove-tailing  these  triangles  together  we  can  form  a  rectangular 
shaped  figure  whose  length  is  15.81,  and  whose  width  is  5,  so  that 
the  area  equals  5  x  15.81,  or  79.05.    Hence,  in  modern  notation, 
-  x  diameter  is  the  area. 

4 

In  the  1660  edition  of  the  Ketsugi-sJw  he  gives  the  surface 
of  a  sphere  as  one-fourth  the  square  of  its  circumference,  which 


V.  The  Third  Period. 


75 


would  make  it  n2;-2  instead  of  4Trr2.  In  the  1684  edition,1 
however,  he  says  that  this  is  incorrect,  although  he  asserts 
that  it  had  been  stated  by  Mori,  Yoshida,  Imamura,  Takahara, 
Hiraga,  Shimada,  and  others.  It  seems  that  the  rule  had  been 
derived  from  considering  the  surface  of  the  sphere  as  if  it  were 


20 


36 


34 


63 


,31 


53 


12 


44 


13 


52 


29 


40 


60 


43 


11 


27 


'23 


39 


58 


59 


55 


26 


51s 


14 


19 


35 


30 


32 


'48 


64l 


17 


,57 


16 


56 


the  skin  of  an  orange  that  could  be  removed  and  cut  into 
triangular  forms  and  fitted  together  in  the  same  manner  as 
the  sectors  of  a  circle.  The  error  arose  from  not  considering 
the  curvature  of  the  surface.  To  rectify  the  error  Isomura 


1  Book  IV,  note. 


76  V.  The  Third  Period. 

took  two  concentric  spheres  with  diameters  10  and  10.0002. 
He  then  took  the  differences  of  their  volumes  and  divided  this 
by  o.oooi,  the  thickness  of  the  rind  that  lay  between  the  two 
surfaces.  This  gave  for  the  spherical  surface  314.160000041888, 

from  which  he  deduced  the  formula,  s  =  -~  =  •nd2.  This  in- 
genious process  of  finding  s,  which  of  course  presupposes  the 
ability  to  find  the  volume  of  a  sphere,  has  since  been  employed 
by  several  writers. I 

It  should  be  mentioned,  before  leaving  the  works  of  Isomura, 
that  the  1684  edition  of  the  Ketsugi-sho  contains  a  few  notes 
in  which  an  attempt  is  made  to  solve  some  simultaneous  linear 
equations  by  the  method  of  the  "Celestial  element"  already 
described.  The  author  states,  however,  that  he  does  not  favor 
this  method,  since  it  seems  to  fetter  the  mind,  the  older 
arithmetical  methods  being  preferable. 

Isomura  seems  not  to  have  placed  in  his  writings  all  of  his 
knowledge  of  such  subjects  as  the  circle,  for  he  distinctly 
states  that  one  must  be  personally  instructed  in  regard  to  some 
of  these  measures.  Possibly  he  was  desirous  of  keeping  this 
knowledge  a  secret,  in  the  same  way  that  Tartaglia  wished  to 
keep  his  solution  of  the  cubic.  Indeed,  there  is  a  igth  century 
manuscript  that  is  anonymous,  although  probably  written  by 
Furukawa  Ken,  bearing  the  title  Sanwa  Zuihitsu  (Miscellany 
about  Mathematical  Subjects),  in  which  it  is  related  that  Iso- 
mura possessed  a  secret  book  upon  the  mensuration  of  the 
circle,  and  in  particular  upon  the  circular  arc.  It  is  said  that 
this  was  later  owned  by  Watanabe  Manzo  Kazu,  one  of  Aida 
Ammei's  pupils,  and  a  retainer  of  the  Lord  of  Nihommatsu, 
where  Isomura  one  time  dwelt.  The  writer  of  the  Sanwa 
Zuihitsu  asserts  that  he  saw  the  book  in  1811,  during  a  visit 
at  his  home  by  Watanabe,  and  that  he  made  a  copy  of  it  at 
that  time.  He  says  that  the  methods  were  not  modern  and 
that  they  contained  fallacies,  but  that  the  explanations  were 

1  It  is  given  in  Takebe  Kenko's  manuscript  work,  the  Fnkyii  Tetsujtitsu 
of  1722,  in  an  anonymous  manuscript  entitled  Kigenkai,  and  in  a  work  of 
the  I gth  century  by  Wada  Nei. 


V.  The  Third  Period.  77 

minute.  The  title  of  the  work  was  Koshigen  Yensetsu  Hompo, 
and  it  was  dated  the  i$th  day  of  the  3d  month  of  1679. 

Next  in  rank  to  Isomura,  in  this  period,  was  Muramatsu 
Kudayu  Mosei.1  He  was  a  pupil  of  Hiraga  Yasuhide,  a 
distinguished  teacher  but  not  a  writer,  who  served  under  the 
feudal  Lord  of  Mito,  meeting  with  a  tragic  death  in  1683. 2 

Muramatsu  was  a  retainer  of  Asano,  Lord  of  Ako,  whose 
forced  suicide  caused  the  heroic  deed  of  the  "Forty -seven 
Ronins"  so  familiar  to  readers  of  Japanese  annals.  Muramatsu 
is  sometimes  recorded  as  one  of  the  honored  "Forty-seven", 
but  it  was  his  adopted  son,  Kihei,  and  Kihei's  son,  who  were 
among  the  number.  3  As  to  Muramatsu  himself,  he  died  at 
an  advanced  age  after  a  life  of  great  activity  in  his  chosen 
field. 

In  1663  Muramatsu  began  the  publication  of  a  work  in  five 
books,  entitled  the  Sanso.*  In  this  he  treats  chiefly  of  arith- 
metic and  mensuration,  following  in  part  the  Chinese  work, 
Suan-hsiao  Chi-meng,  written  by  Chu  Chi-chieh,  as  mentioned 
on  page  48,  but  he  fails  to  introduce  the  method  of  the  "Ce- 
lestial element".  The  most  noteworthy  part  of  his  work  relates 
to  the  study  of  polygons  s  and  to  the  mensuration  of  the  circle. 6 

Taking  the  radius  of  the  circumscribed  circle  as  5,  he  cal- 
culates the  sides  of  the  regular  polygons  as  follows: 

No.  of  sides.     Length  of  side.     No.  of  sides.     Length  of  side. 


5 

5.8778 

ii 

2.801586 

6 

5 

12 

2.5875 

7 

4-3506 

13 

2-393 

8 

3.82682 

14 

2.22678 

9 

3.4102 

15 

2.07953 

IO 

3.0876 

16 

1.95093 

1  Not  Matsumura,  as  given  by  ENDO.    The  name  Mosei  appears  as  Shigekiyo 
in  his  Mantoku  Jinko-ki  (1665). 

2  See  the  Stories  told  by  Araki. 

3  AOYAMA,  Lives  of  the  Forty-seven  Loyal  Men  (in  Japanese). 

4  The  last  book  bears  the  date  1684,  and  may  not  have  appeared  earlier. 

5  Book  2.  6  Book  4. 


78  V.  The  Third  Period. 

To  calculate  the  circumference  Muramatsu  begins  with  an 
inscribed  square  whose  diagonal  is  unity.  He  then  doubles  the 
number  of  sides,  forming  a  regular  octagon,  the  diameter  of 
the  circumscribed  circle  being  one.  He  continues  to  double 
the  number  of  sides  until  a  regular  inscribed  polygon  of  3278 
sides  is  reached.  He  computes  the  perimeters  of  these  sides 
in  order,  by  applying  the  Pythagorean  Theorem,  with  the 
following  results: 

No.  of  sides.  Perimeter. 

2*  3.06146745892071817384 

24  3.121445152258052370213 


26  3.140331156954753 

2?  3.1412792509327729134016 

28  3.141513801144301128448 

29  3.14157294036/091435162 
210  3.14158772527715976659 
2"  3.141591421511186733296 
212  3.1415923455701046761472 
2*3  3.1415925765848605108681 
214  3.14159263433855298 

2*5  3.141592648777698869248 

Having  reached  this  point,  Muramatsu  proceeded  to  compare 
the  various  Chinese  values  of  TT,  and  stated  his  conclusion  that 
3.14  should  be  taken,  unaware  of  the  fact  that  he  had  found 
the  first  8  figures  correctly.1 

Muramatsu  gives  a  brief  statement  as  to  his  method  of 
finding  the  volume  of  a  sphere,  but  does  not  enter  into  details.2 
He  takes  10  as  the  diameter,  and  by  means  of  parallel  planes 
he  cuts  the  sphere  into  100  segments  of  equal  altitude.  He 
then  assumes  that  each  of  these  segments  is  a  cylinder,  either 
with  the  greater  of  the  two  bases  as  its  base,  or  with  the 
lesser  one.  If  he  takes  the  greater  base,  the  sum  of  the  vol- 

1  EXDO,  Book  I,  p.  70. 

2  The  Sanso,  Book  5. 


V.  The  Third  Period. 


79 


umes  is  562.5  cubic  units;  but  if  he  takes  the  lesser  one  this 
sum  is  only  493.04  cubic  units.  He  then  says  that  the  volume 
of  the  sphere  lies  between  these  limits,  and  he  assumes,  without, 


Fig.  24.     Magic  circle,  from  Muramatsu  Kudayii  Mosei's  Mantoku  Jinko-ki  (1665). 

stating  his  reasons,  that  it  is  524,  which  is  somewhat  less  than 
either  their  arithmetic  (527)  or  their  geometric  (526.6)  mean,1 
and  which  is  equivalent  to  taking  TT  as  3.144. 

Muramatsu  was  also  interested  in  magic  squares 2  and  magic 

*  ENDO  thinks  that  he  may  have  reached  this  value  by  cutting  the  sphere 
into  200,  400  or  some  other  number  of  equal  parts.     History,  Book  I,  p.  Jl. 
2  His  rakusho  (afterwards  called  hojiri)  problems. 


8o 


V.  The  Third  Period. 


circles.1  One  of  his  magic  squares  has  19*  cells,  as  did  one 
published  by  Nozawa  Teicho  in  the  following  year.8  One  of 
his  magic  circles,  in  which  129  numbers  are  used,  is  shown  in 
Fig.  24  on  page  79.  With  the  numbers  expressed  in  Arabic 
numerals  it  is  as  follows: 


In  Muramatsu's  work  also  appears  a  variant  of  the  famous 
old  Josephus  problem,  as  it  is  often  called  in  the  West,  a 
problem  that  had  already  appeared  in  the  Jinko-ki  of  Yoshida. 


1  His  ensan  problems.     Sanso,  Book  2. 

2  In  his  Dokai-sho  of  1664, 


V.  The  Third  Period. 


81 


Fig.  25.     The  Josephus    problem,   from   Muramatsu  Kudayu   Mosei's  Mantoku    /v\ 

Jinko-ri  (1665). 


82 


V.  The  Third  Period. 


As  given  by  Seki,  a  little  later,  it  is  as  follows:  "Once  upon 
a  time  there  lived  a  wealthy  farmer  who  had  thirty  children, 
half  being  born  of  his  first  wife  and  half  of  his  second  one. 
The  latter  wished  a  favorite  son  to  inherit  all  the  property, 
and  accordingly  she  asked  him  one  day,  saying:  Would  it 
not  be  well  to  arrange  our  thirty  children  on  a  circle,  calling 


Fig.  26.     The  Josephus  problem,   from  Miyake  Kenryfi's  Shojutsu 
Sangaku  Zuye  (1795  edition). 

one  of  them  the  first  and  counting  out  every  tenth  one  until 
there  should  remain  only  one,  who  should  be  called  the  heir. 
The  husband  assenting,  the  wife  arranged  the  children  as  shown 
in  the  figure T.  The  counting  then  began  as  shown  and  resulted 
in  the  elimination  of  fourteen  step-children  at  once,  leaving 
only  one.  Thereupon  the  wife,  feeling  confident  of  her  success, 


1  The  step  children  are  represented  by  dark  circles,  and  her  own  children 
by  light  ones.     In  the  old  manuscripts  the  latter  are  colored  red. 


V.  The  Third  Period. 


said:  Now  that  the  elimination  has  proceeded  to  this  stage, 
let  us  reverse  the  order,  beginning  with  the  child  I  choose. 
The  husband  agreed  again,  and  the  counting  proceeded  in  the 
reverse  order,  with  the  unexpected  result  that  all  of  the  second 
wife's  children  were  stricken  out  and  there  remained  only  the 
step-child,  and  accordingly  he  inherited  the  property."  The 
original  is  shown  in  Fig.  25,  and  an  interesting  illustration  from 
Miyake's  work  of  1795  in  Fig.  26,  but  the  following  diagram 
will  assist  the  reader: 


120     End 


V 


Reverse  count  begins  here 
Figures  outside. 


Direct  count  begins  here 
Figures  inside. 


no 


Perhaps   it  is  more  in  accord  with  oriental  than   with   oc- 
cidental   nature    that    the   interesting    agreement   should    have 

6* 


84  V.  The  Third  Period. 

remained  in  force,  with  the  result  that  the  heir  should  have 
been  a  step-son  of  the  wife  who  planned  the  arrangement. 
Seki  also  gave  the  problem,  having  obtained  it  from  the  Jinko- 
ki  of  Yoshida,  although  he  mentions  only  the  fact  that  it  is  an 
old  tradition.  Possibly  it  was  one  of  Michinori's  problems  in 
the  twelfth  century,  but  whether  it  started  in  the  East  and 
made  its  way  to  the  West,  or  vice  versa,  we  do  not  know. 
The  earliest  definite  trace  of  the  analogous  problem  in  Europe 
is  in  the  Codex  Einsidelensis,  early  in  the  tenth  century, 
although  a  Latin  work  of  Roman  times1  attributes  it  to  Flavius 
Josephus.  It  is  also  mentioned  in  an  eleventh  century  manu- 
script in  Munich  and  in  the  Ta'hbula  of  Rabbi  Abraham  ben  Ezra 
(d.  1067).  It  is  to  the  latter  that  Elias  Levita,  who  seems  first 
to  have  made  it  known  in  print  (1518),  assigns  its  origin.  It 
commonly  appears  as  a  problem  relating  to  Turks  and  Christians, 
or  to  Jews  and  Christians,  half  of  whom  must  be  sacrificed  to 
save  a  sinking  ship.3 

The  next  writer  of  note  was  Nozawa  Teicho,  who  published 
his  Dokai-sho  in  1664,  and  who  followed  the  custom  begun  by 
Yoshida  in  the  proposing  of  problems  for  solution.  Nozawa 
solved  all  of  Isomura's  problems  and  proposed  a  hundred  new 
ones.  He  also  suggested  the  quadrature  of  the  circle  by  cutting 
it  into  a  number  of  segments  and  then  summing  these  partial 
areas.  He  went  so  far  as  to  suggest  the  same  plan  for  the 
sphere,  but  in  neither  case  does  he  carry  his  work  to  com- 
pletion. It  is  of  interest  to  see  this  approach  to  the  calculus 
in  Japan,  contemporary  with  the  like  approach  at  this  time  in 
Europe.  Muramatsu  had  approximated  the  volume  of  the 

*  De  bello  judaico,  III,  16.  This  was  formerly  attributed  to  Hegesippus  of 
the  second  century  A.  D.,  but  it  is  now  thought  to  be  by  a  later  writer  of 
uncertain  date. 

2  Common  names  are  Ludus  Josephi,  Josephsspiel,  Sankt  Peder's  lek  (Swedish), 
and  the  Josephus  Problem.  The  Japanese  name  was  Mameko-date,  the  step- 
children problem.  It  was  very  common  in  early  printed  books  on  arithmetic, 
as  in  those  of  Cardan  (1539),  Ramus  (1569),  and  Thierfelder  (1587).  The  best 
Japanese  commentary  on  the  problem  is  Fujita  Sadusuke's  Sandatsti  Kaigi 
(Commentary  on  Sandatsu),  1774. 


V.  The  Third  Period. 


sphere  by  means  of  the  summation  of  cylinders  formed  on 
circles  cut  by  parallel  planes.  He  had  taken  100  of  these 
sections,  and  possibly  more,  and  had  taken  some  kind  of 
average  that  led  him  to  fix  upon  524  as  the  volume  of  a 
sphere  of  radius  5.  Nozawa  apparently  intends  to  go  a  step 
further  and  to  take  thinner  laminae,  thus  approaching  the 
method  used  by  Cavalieri  in  his  Methodus  indivisibilibus. T  It  is 
possible,  as  we  shall  see  later,  that  some  hint  of  the  methods 
of  the  West  had  already  reached  the  Far  East,  or  it  is  possible 
that,  as  seems  so  often  the  case,  the  world  was  merely  show- 
ing that  it  was  intellectually  maturing  at  about  the  same  rate 
in  regions  far  remote  one  from  the  other. 

Two  years  later,  in  1666,  the  annns  mirabilis  of  England, 
Sato  Seiko2  wrote  his  work  entitled  Kongenki.  In  this  he 
attempted  to  solve  the  problems  proposed  by  Isomura  and 
Nozawa,  and  he  set  forth  150  new  questions.  Mention  should 
also  be  made  of  his  interest  in  magic  circles.  Since  with  him 
closes  the  attempts  at  the  mensuration  of  the  circle  and  sphere 
prior  to  the  work  of  Seki,  it  is  proper  to  give  in  tabular  form 
the  results  up  to  this  time.^ 


Author 

Date 

it 

Area  of  Circle 

Volume  of 
sphere 

Yoshida 

1627 

3-16 

0.79 

0.5625 

Imamura 

1639 

3.162 

0.7905 

0.51 

Yamada                  1656 

3.162 

0.7905 

0.4934 

Shibamura 

1657 

3.162 

0.7905 

0.525 

Isomura 

1660 

3.162 

0.7905 

0.51 

Muramatsu             1663 

3-14 

0.785 

0.524 

Nozawa 

1664 

3-14 

0.785 

0.523 

Sato 

1666 

3-14 

0.785 

0.519 

1  Written  in  1629,  but  printed  in  1635. 

2  Given  incorrectly  in  FUKUDA'S  Sampo  Tamatebako  of  1879,  and  in  ENDO, 
Book  I,  p.  73,  as  Sato  Seioku. 

3  The  table  in  substantially  this  form  appears  in  HAYASHI'S  History,  p. -37. 
See  also  HERZER,  P.,  loc.  cit.,  p.  35  of  the  Kiel  reprint  of  1905 ;  ENDO,  I,  p.  75. 


86  V.  The  Third  Period. 

Sato's  Kongenki  of  1666  is  particularly  noteworthy  as  being 
the  first  Japanese  treatise  in  which  the  "Celestial  element" 
method  in  algebra,  as  set  forth  in  the  Suan-hsiao  Chi-meng^ 
is  successfully  used.  Some  of  the  problems  given  by  him 
require  the  solution  of  numerical  equations  of  degree  as  high 
as  the  sixth,  and  it  is  here  that  Sato  shows  his  advance  over 
his  predecessors.  The  numerical  quadratic  had  been  solved  in 
Japan  before  his  time,  and  even  certain  numerical  cubics,  but 
Sato  was  the  first  to  carry  this  method  of  solution  to  equa- 
tions of  higher  degree.  In  spite  of  the  fact  that  he  knew  the 
principle,  Sato  showed  little  desire  to  carry  it  out,  however, 
so  that  it  was  left  to  his  successor  to  make  more  widely  known 
the  Chinese  method  and  to  show  its  great  possibilities. 

This  successor  was  Sawaguchi  Kazuyuki,2  a  pupil  of  Taka- 
hara  Kisshu,  and  afterwards  a  pupil  of  the  great  Seki.  In  1 670 
Sawaguchi  wrote  the  Kokon  Sampo-ki,  the  "Old  and  New  Me- 
thods of  Mathematics".  The  work  consists  of  seven  books,  the 
first  three  of  which  contain  the  ordinary  mathematical  work 
of  the  time,  and  the  next  three  a  solution  by  means  of  equa- 
tions of  the  problems  proposed  by  Sato.  3  He  also  followed 
Nozawa  in  attempting  to  use  a  crude  calculus  (Fig.  27)  some- 
what like  that  known  to  Cavalieri.  Sawaguchi  was  for  a  time 
a  retainer  of  Lord  Seki  Bingo-no-Kami,  but  through  some  fault 
of  his  own  he  lost  the  position  and  the  closing  years  of  his 
life  were  spent  in  obscurity.4 

Sawaguchi's  solutions  of  Sato's  problems  are  not  given  in 
full.  The  equations  are  stated,  but  these  are  followed  by  the 
answers  only.  An  equation  of  the  first  degree  is  called  a 
kijo  shiki,  "divisional  expression",  inasmuch  as  only  division  is 
needed  in  its  solution,  of  course  after  the  transposition  and 

1  See  p.  48. 

2  In  later  years  he  seems,  according  to  the  Stories  told  by  Araki,  to  have 
changed  his  name  to  Goto  Kakubei,  although  other  writers  take  the  two  to 
be  distinct  personages. 

3  It  should  also  be  mentioned  that  a  similar  use  of  equations  is  found  in 
Sugiyama  Teiji's  work  that  appeared  in  the  same  year. 

4  The  Stories  told  by  Araki. 


V.  The  Third  Period.  87 

uniting  of  terms.  Equations  of  higher  degree  are  called  kaiho 
shiki,  "root-extracting  expressions".  As  a  rule  only  a  single 
root  of  an  equation  is  taken,  although  in  a  few  problems  this 
rule  is  not  followed.1  This  idea  of  the  plurality  of  roots  is  a 


m 


tv 

A* 


90 


-tz 
?A 


if* 


£ftg 


1311 


lip 


^LSI5I5I 


t 


Fig.  27.     Early   steps  in   the   calculus.     From   Sawaguchi   Kazuyuki's   Kokon 

Sampo-ki  (1670). 

noteworthy  advance  upon  the  work  of  the  earlier  Chinese 
writers,  since  the  latter  had  recognized  only  one  root  to  any 
equation.  As  is  usual  in  such  forward  movements,  however, 
Sawaguchi  did  not  recognize  the  significance  of  the  plural 


1  Sato  had  already  recognised  the  plurality  of  roots  in  his  Kongenki. 


88  V.  The  Third  Period. 

roots,  calling  problems  which  yielded  them  erroneous  in  their 
nature. 

That  Sawaguchi's  methods  may  be  understood  as  fully  as 
the  nature  of  his  work  allows,  a  few  of  his  solutions  of  Sato's 
problems  are  set  forth: 

"There  is  a  right  triangle  whose  hypotenuse  is  6,  and  the 
sum  of  whose  area  and  the  square  root  of  one  side  is  7.2384. 
Required  the  other  two  sides".  (No.  64.) 

Sawaguchi  gives  the  following  direstions: 

"Take  the  'Celestial  element'  to  be  the  first  side.  Square 
this  and  subtract  the  result  from  the  square  of  the  hypotenuse. 
The  remainder  is  the  square  of  the  second  side.  Multiplying 
this  by  the  square  of  the  first  side,  we  have  4  times  the  square 
of  the  area,  which  will  be  called  A.  Let  4  times  the  square  of  the 
first  side  be  called  B.  Arrange  the  sum,  square  it,  and  multiply 
by  4.  From  the  result  subtract  A  and  B.  The  square  of  the 
remainder  is  4  times  the  product  of  A  and  B,  and  this  we 
shall  call  X.  Arrange  A,  multiply  by  B,  take  4  times  the 
product,  and  subtract  the  quantity  from  X,  thus  obtaining  an 
equation  of  the  8th  degree.  This  gives,  evolved  in  the  reverse 
method, *  the  first  side."  The  result  for  the  two  sides  are  then 
given  as  5.76,  and  I.68.2 

Sato's  problem  No.  16  is  as  follows:  "There  is  a  circle  from 
within  which  a  square  is  cut,  the  remaining  portion  having  an 
area  of  47.6255.  If  the  diameter  of  the  circle  is  7  more  than 
the  square  root  of  a  side  of  the  square,  it  is  required  to  find 
the  diameter  of  the  circle  and  the  side  of  the  square."  -5  Sawa- 
guchi looks  upon  the  problem  as  "deranged",  since  it  has  two 
solutions,  viz.,  d=c>,  s  =  4,  and  ^=7.8242133...  and  s  = 
0.67932764 ....  He  therefore  changes  the  quantities  as  given  in 


1  That  is,  when   the   signs   of  the  coefficients   are  changed  in  the  course 
of  the  operation. 

2  Expressed  in  modern   symbols,   let  j  =  the  sum,  7.2384,   ^==the  hypo- 
tenuse, and  ^-  =  the  first  side.    Then,  by  his  rule,  [4^  —  (/i2  —  x2)  x2 — 4*2]2 

16*4    (fc  —  X2)  =  o. 

3  I.  e.,  —  u  d*  —  jz  =  47 . 6255 ,  and   d  —  Vs  =  7. 


V.  The  Third  Period. 


89 


the  problem,  making  the  area  12.278,  and  the  difference  4.  He 
then  considers  the  equation  as  before,  viz.,  —  ltd*  —  s2  =  12.278, 
and  d — Vs  =  4.  Then  d  =  6  and  s  =  4,  taking  -  TT  to  be  0.785  5. 

Sawaguchi  next  considers  a  problem  from  the  Dokai-sho  of 
Nozawa  Teicho  (1664),  viz:  "There  is  a  rectangular  piece  of 
land  300  measures  long  and  132  measures  wide.  It  is  to  be 
equally  divided  among  4  men  as  here  shown,  in  such  manner 


that  three  of  the  portions  shall  be  squares.     Required  the  di- 
mensions of  the  parts." 

Sato  gives  two  solutions  of  this  problem  in  his  Kongenki,  as 
follows: 

1.  Each   of  the  square  portions  is  90  measures  on  a  side; 
the   fourth   portion   is    27  measures  wide;   and  the   roads  are 
each  15  measures  wide. 

2.  Each  of  the  square  portions  is  60  measures  on  a  side; 
the  fourth  portion  is  12  measures  wide;  and  the  roads  are  each 
60  measures  wide. 

This  solution  of  Sato's  leads  Sawaguchi  to  dilate  upon  the 
subtle  nature  of  mathematics  that  permits  of  more  than  one 
solution  to  a  problem  that  is  apparently  simple. 

Of  the  hundred  and  fifty  problems  in  Sato's  work  Sawa- 
guchi says  that  he  leaves  some  sixteen  unsolved  because  they 
relate  to  the  circle.  He  announces,  however,  that  it  is  his  in- 
tention to  consider  problems  of  this  nature  orally  with  his 
pupils,  and  he  gives  without  explanation  the  value  of  TT  as 
3.142. 

Two  of  the  sixteen  unsolved  problems  are  as  follows: — 


9O  V.  The  Third  Period. 

"The  area  of  a  sector  of  a  circle  is  41.3112,  the  radius  is 
8.5,  and  the  altitude  of  the  segment  cut  off  by  a  chord  is  2. 
Required  to  find  the  chord."  (No.  34.) 

"From  a  segment  of  a  circle  a  circle  is  cut  out,  leaving  the 
remaining  area  97.27632.  The  chord  is  24,  and  the  two  parts 


of  the  altitude,  after  the  circle  cuts  out  a  portion  as  shown  in 
the  figure,  are  each  1.8.  Required  the  diameter  of  the  small 
circle." 

The  seventh  and  last  book  of  Sawaguchi's  work  consists  of 
fifteen  new  problems,  all  of  which  were  solved  four  years  later 
by  Seki,  who  states  that  one  of  them  leads  to  an  equation  of 
the  1458th  degree.  This  equation  was  substantially  solved 
twenty  years  later  by  Miyagi  Seiko  of  Kyoto,  in  his  work 
entitled  Wakan  Sampo. 


CHAPTER  VI. 
Seki  Kowa. 

In  the  third  month  according  to  the  lunar  calendar,  in  the 
year  1642  of  our  era,  a  son  was  born  to  Uchiyama  Shichibei, 
a  member  of  the  samurai  class  living  at  Fujioka  in  the  pro- 
vince of  Kozuke.  *  While  still  in  his  infancy  this  child,  a 
younger  son  of  his  parents,  was  adopted  into  another  noble 
family,  that  of  Seki  Gorozayemon,  and  hence  there  was  given 
to  him  the  name  of  Seki  by  which  he  is  commonly  known  to 
the  world.  Seki  Shinsuke  Kowa2  was  born  in  the  same  year 3 
in  which  Galileo  died,  and  at  a  time  of  great  activity  in  the 
mathematical  world  both  of  the  East  and  the  West.  And  just 
as  Newton,  in  considering  the  labors  of  such  of  his  immediate 
predecessors  as  Kepler,  Cavalieri,  Descartes,  Fermat,  and  Barrow, 
was  able  to  say  that  he  had  stood  upon  the  shoulders  of  giants, 
so  Seki  came  at  an  auspicious  time  for  a  great  mathematical 
advance  in  Japan,  with  the  labors  of  Yoshida,  Imamura,  Iso- 
mura,  Muramatsu,  and  Sawaguchi  upon  which  to  build.  The 
coincidence  of  birth  seems  all  the  more  significant  because  of 
the  possible  similarity  of  achievement,  Newton  having  invented 
the  calculus  of  fluxions  in  the  West,  while  Seki  possibly 
invented  the  yenri  or  "circle  principle"  in  the  East,  each 

1  Not  far  from  Yedo,  the  Shogun's  capital,  the  present  TokyS. 

2  Or  Takakazu.     On  the  life  of  Seki  see  MIKAMI,  Y.,  Seki  and  Shibukawa, 
Jahresbericht    der    Deuischen    Mathematiker-Vereinigung,     Vol.  XVII,     p.    187; 
ENDO,   Book  II,   p.    40;    OZAWA,    Lineage   of   Mathematicians    (in    Japanese) ; 
HAYASHI,  History,  part  I,  p.  43,  and  the  memorial  volume  (in  Japanese)  issued 
on  the  two-hundredth  anniversary  of  Seki's  death,   1908. 

3  C.  KAWAKITA,  in   an   article  in   the  Honcho  Sugaku   Koenshu,   says  that 
some  believe  Seki  to  have  been  born  in  1637. 


92  VI.  Seki  Kowa. 

designed  to  accomplish  much  the  same  purpose,  and  each 
destined  to  material  improvement  in  later  generations.  The 
yenri  is  not  any  too  well  known  and  it  is  somewhat  difficult 
to  judge  of  its  comparative  value,  Japanese  scholars  themselves 
being  undecided  as  to  the  relative  merits  of  this  form  of  the 
calculus  and  that  given  to  the  world  by  Newton  and  Leibnitz. * 

Seki's  great  abilities  showed  themselves  at  an  early  age. 
The  story  goes  that  when  he  was  only  five  he  pointed  out 
the  errors  of  his  elders  in  certain  calculations  which  were  being 
discussed  in  his  presence,  and  that  the  people  so  marveled  at 
his  attainments  that  they  gave  him  the  title  of  divine  child.2 

Another  story  relates  that  when  he  was  but  nine  years  of 
age,  Seki  one  time  saw  a  servant  studying  the  Jinko-ki  of 
Yoshida.  And  when  the  servant  was  perplexed  over  a  certain 
problem,  Seki  volunteered  to  help  him,  and  easily  showed  him 
the  proper  solution.  3  This  second  story  varies  with  the  narrator, 
Kamizawa  Teikan4  telling  us  that  the  servant  first  interested 
the  youthful  Seki  in  the  arithmetic  of  the  Jinko-ki,  and  then 
taught  him  his  first  mathematics.  Others  s  say  that  Seki 
learned  mathematics  from  the  great  teacher  Takahara  Kisshu 
who,  it  will  be  remembered,  had  sat  at  the  feet  of  Mori  as 
one  of  his  san-shi,  although  this  belief  is  not  generally  held. 
Most  writers6  agree  that  he  was  self-made  and  self-educated, 

1  Thus  ENDO  feels  that  the  yenri  was  quite  equal  to  the  calculus  (History, 
Book  III,  p.  203).     See  also  HAYASHI,  History,  part  I,   p.  44,  and  the  Honcho 
Siigaku  Kdenshit,  pp.  33 — 36.     Opposed  to  this  idea  is  Professor  R.  FUJISAWA 
of  the  University  of  Tokyo  who  asserts  that  the  yenri  resembles  the  Chinese 
methods    and   is   much   inferior   to  the  calculus.     The  question  will  be  more 
fully  considered  in  a  later  chapter. 

2  KAMIZAWA    TEIKAV  (1710 — 1795),    Okinagusa,    Book  VIII.       KAMIZAWA 
lived  at  KyQto.    This  title  was  also  placed  upon  the  monument  to  Seki  erected 
in  Tokyo  in  1794. 

3  Kuichi  Sanjin,  in  the  Sugaku  Hochi,  No.  55. 

4  Okinagusa,  Book  VIII. 

5  See  FUKUDA'S   Sampo  Tamatebako,  1879;  ENDO,  Book  II,  p.  40;  HAYASHI 
in  the  Honcho  Sugaku  Koenshu,  1908. 

6  Fujita  Sadasuke   in   the  preface  to  his  Seiyo  Sampo,  17795   Ozawa  Seiyo 
in  his  Lineage  of  Mathematicians   (in  Japanese),   1801;   the   anonymous   manu- 
script entitled  Sanka  Keizu. 


VI.  Seki  Kowa.  93 

his  works  showing  no  apparent  influence  of  other  teachers,  but 
on  the  contrary  displaying  an  originality  that  may  well  have 
led  him  to  instruct  himself  from  his  youth  up. T  Whatever 
may  have  been  his  early  training  Seki  must  have  progressed 
very  rapidly,  for  he  early  acquired  a  library  of  the  standard 
Japanese  and  Chinese  works  on  mathematics,  and  learned, 
apparently  from  the  Suan-hsiao  Cki-meng,2  the  method  of 
solving  the  numerical  higher  equation.  And  with  this  progress 
in  learning  came  a  popular  appreciation  that  soon  surrounded 
him  with  pupils  and  that  gave  to  him  the  title  of  The  Arith- 
metical Sage. 3  In  due  time  he,  as  a  descendent  of  the  samurai 
class,  served  in  public  capacity,  his  office  being  that  of  ex- 
aminer  of  accounts  to  the  Lord  of  Koshu,  just  as  Newton 
^became  master  of  the  mint  under  Queen  Anne.  When  his 
lord  became  heir  to  the  Shogun,  Seki  became  a  Shogunate 
samurai,  and  in  1704  was  given  a  position  of  honor  as  master 
of  ceremonies  in  the  Shogun's  household.4  He  died  on  the 
24th  day  of  the  loth  month  in  the  year  1708,  at  the  age  of 
sixty-six,  leaving  no  descendents  of  his  own  blood,  s  He  was 
buried  in  a  Buddhist  cemetery,  the  Jorinji,  at  Ushigome  in 
Yedo  (Tokyo),  where  eighty  years  later  his  tomb  was  rebuilt, 
as  the  inscription  tell  us,  by  mathematicians  of  his  school. 

Several  stories  are  told  of  Seki,  some  of  which  throw  interest- 
ing sides  lights  upon  his  character. 6  One  of  these  relates  that 
he  one  time  journeyed  from  Yedo  to  Kofu,  a  city  in  Koshu, 
or  the  Province  of  Kai,  on  a  mission  from  his  lord.  Traveling 
in  a  palanquin  he  amused  himself  by  noting  the  directions  and 

1  The   fact  that  the  long   epitaph   upon  his   tomb  makes   no  mention  of 
any  teacher  points  to  the  same  conclusion. 

2  In  the   Okinagusa  of  Kamizawa  this  is  given  as  the  Sangaku  Gomo,  but  in 
an  anonymous  manuscript  entitled  the  Samoa  Zuihitsu  the  Chinese  classic  is 
specially  given  on  the  authority  of  one  Saito  in  his  Burin  Inken  Roku. 

3  In  Japanese,  Sansei.     This  title  was  also  carved  upon  his  tomb. 

4  KAMIZAWA,  Okinagusa,   Book  VIII;    Kuichi  Sanjin   in  the  Stlgaku   Hochi, 
No.  55;  ENDO,  Book  II,  p.  40. 

5  His   heir   was   Shinshichi,   or   ShinshichirO,  a  nephew.     ENDO,  Book  II, 
p.  81. 

6  KAMIZAWA,  Okinagusa,  Book  VIII. 


94  VI.   Seki  Kowa. 

distances,  the  objects  along  the  way,  the  elevations  and  de- 
pressions, and  all  that  characterized  the  topography  of  the 
region,  jotting  down  the  results  upon  paper  as  he  went.  From 
these  notes  he  prepared  a  map  of  the  region  so  minutely  and 
carefully  drawn  that  on  his  return  to  Yedo  his  master  was 
greatly  impressed  with  the  powers  of  description  of  one  who 
traveled  like  a  samurai  but  observed  like  a  geographer. 

Another  story  relates  how  the  Shogun,  who  had  been  the 
Lord  of  Koshu,  once  upon  a  time  decided  to  distribute  equal 
portions  of  a  large  piece  of  precious  incense  wood  among  the 
members  of  his  family.  But  when  the  official  who  was  to  cut 
the  wood  attempted  the  division  he  found  no  way  of  meeting 
his  lord's  demand  that  the  shares  should  be  equal.  He  there- 
fore appealed  to  his  brother  officials  who  with  one  accord, 
advised  him  that  no  one  could  determine  the  method  of  cutting 
the  precious  wood  save  only  Seki.  Much  relieved,  the  official 
appealed  to  "The  Arithmetical  Sage"  and  not  in  vain.1 

It  is  also  told  of  Seki  that  a  wonderful  clock  was  sent  from 
the  Emperor  of  China  as  a  present  to  the  Shogun,  so  arranged 
that  the  figure  of  a  man  would  strike  the  hours.  And  after 
some  years  a  delicate  spring  became  deranged,  so  that  the 
figure  would  no  longer  strike  the  bell.  Then  were  called  in 
the  most  skilful  artisans  of  the  land,  but  none  was  able  to 
repair  the  clock,  until  Seki  heard  of  his  master's  trouble.  Asking 
that  he  might  take  the  clock  to  his  own  home,  he  soon  restored 
it  to  the  Shogun  successfully  repaired  and  again  correctly 
striking  the  hours. 

Such  anecdotes  have  some  value  in  showing  the  acumen 
and  versatility  of  the  man,  and  they  explain  why  he  should 
have  been  sought  for  a  post  of  such  responsibility  as  that  of 
examiner  of  accounts.2 

The  name  of  Seki  has  long  been  associated  with  the  yenri, 
a  form  of  ihe  calculus  that  was  possibly  invented  by  him,  and 

1  The  story  is  evidently  based  upon  the  problem  of  Yoshida  already  given 
on  page  66. 

2  KAMIZAWA,  Okinagusa,  Book  VIII. 


VI.  Seki  Kovva.  95 

that  will  be  considered  in  Qiapter  VIII.)  It  is  with  greater 
certainty  that  he  is  known  for  msTJenzan  method,  an  algebraic 
system  that  improved  upon  the  method  of  the  "Celestial  ele- 
ment" inherited  from  the  Chinese;  for  the  Yendan  jutsu,  a 
scheme  by  which  the  treatment  of  equations  and  other  branches 
of  algebra  is  simpler  than  by  the  methods  inherited  from  China 
and  improved  by  such  Japanese  writers  as  Isomura  and  Sawa- 
guchi,  and  for  his  work  in  determinants  that  antedated  what 
has  heretofore  been  considered  the  first  discovery,  namely  the 
investigations  of  Leibnitz. 

As  to  his  works,  it  is  said  that  he  left  hundreds  of  un- 
published manuscripts,1  but  if  this  be  true  most  of  them  are 
lost2  He  also  published  the  Hatsubi Sampo  in  1674.3  In  this 
he  solved  the  fifteen  problems  given  in  Sawaguchi's  Kokon 
Sampo-ki  of  1670,  only  the  final  equations  being  given.4 

As  to  Seki's  real  power,  and  as  to  the  justice  of  ranking  him 
with  his  great  contemporaries  of  the  West,  there  is  much  doubt. 
He  certainly  improved  the  methods  used  in  algebra,  but  we 
are  Jnot  at  all  sure  that  his  name  is  properly  connected  with 
the  yenri. 

For  this  reason,  and  because  of  his  fame,  it  has  been  thought 
best  to  enter  more  fully  into  his  work  than  into  that  of  any 
of  his  predecessors,  so  that  the  reader  may  have  before  him 
the  material  for  independent  judgment. 

First  it  is  proposed  to  set  forth  a  few  of  the  problems  that 
were  set  by  Sawaguchi,  with  Seki's  equations  and  with  one  of 
Takebe's  solutions. 


1  ENDO,  Book  II,  p.  41. 

2  For  further  particulars  see  ENDO,  loc.  cit.,  and  the  Seki  memorial  volume 
(Seki-ryil  Shichibusho,   or   Seven   Books   on   Mathematics   of  the  Seki  School) 
published  in  Tokyo  in  1908. 

3  This  is  the  work  mentioned  by  Professor   Hayashi   as  the  Hakki  JSampo 
of  Mitaki  and  Mie  (Miye). 

4  In  1685  one  of  Seki's  pupils,  Takebe  Kenko,  published  a  work  entitled 
Hatsubi  Sampo  Yendan  Genkai,  or  the  "Full  explanations  of  the  Hatsubi  Sampo," 
in  which  the  problems  are  explained.    He  states  that  the  blocks  for  printing 
the    work  were   burned  in    1680   and   that   he   had  attempted  to  make  good 
their  loss. 


96  VI.    Seki  Kowa. 

Sawaguchi's  first  problem  is  as  follows:  "In  a  circle  three 
other  circles  are  inscribed  as  here  shown,  the  remaining  area 
being  120  square  units.  The  common  diameter  of  the  two 
smallest  circles  is  5  units  less  than  the  diameter  of  the  one 
that  is  next  in  size.  Required  to  compute  the  diameters  of 
the  various  circles." 


Seki   solves  the  problem  as  follows:  "Arrange  the  'celestial 
element',  taking  it  as  the  diameter  of  the  smallest  circles.    Add 
to  this  the  given  quantity  and  the  result  is  the  diameter  of  the 
middle  circle.     Square  this  and  call  the  result  A. 
6  "Take  twice   the    square    of  the    diameter    of  the   smallest 
circles  and  add  this  to  A,  multiplying  the  sum  by  the  moment 
of  the  circumference.1     Call  this  product  B. 
^  "Multiply  4  times   the    remaining    area   by  the   moment   of 
diameter. 2 

£  "This  being  added  to  B  the  result  is  the  product  of  the 
square  of  the  diameter  of  the  largest  circle  multiplied  by  the 
moment  of  circumference.  This  is  called  C. 3 

1  By'  the  "moment  of  the  circumference''  is   meant   the  numerator  of  the 
fractional   value   of  IT.     This   is   22  in   case  IT  is  taken  as  — . 

2  "Moment   of  diameter'"   means   the   denominator  of  the  fractional  value 
of  IT.     In  the   case   of  — ,  this  is  7.     That  is,  we  have  7x120. 

3  Thus  far  the  solution  is  as  follows:  Let  x  =  the  diameter  of  the  smallest 
circle,  and  y  —  the  diameter  of  the  largest  circle.    Then  x  -f-  5  is  the  diameter 
of  the  so-called  "middle  circle." 


VI.  Seki  Kowa.  97 

"Take  the  diameter  of  the  smallest  circle  and  multiply  it  by 
A  and  by  the  moment  of  the  circumference.  Call  the  result  D.  * 
^"From  four  times  the  diameter  of  the  middle  circle  take  the 
diameter  of  the  smallest  circle,  and  from  C  times  this  product 
take  D.  The  square  of  the  remainder  is  the  product  of  the 
square  of  the  sum  of  four  times  the  diameter  of  the  middle 
circle  and  twice  the  diameter  of  the  smallest  circle,  the  square 
of  the  diameter  of  the  middle  circle,  the  square  of  the  moment 
of  circumference,  and  the  square  of  the  diameter  of  the  largest 
circle.  Call  this  X.2 

O  "The  sum  of  four  times  the  diameter  of  the  middle  circle 
and  twice  the  diameter  of  the  smallest  circle  being  squared, 
multiply  it  by  A  and  by  C  and  by  the  moment  of  circum- 
ference. 3  This  quantity  being  canceled  with  X  we  get  an 
equation  of  the  6th  degree.4  Finding  the  root  of  this  equation 
according  to  the  reversed  orders  we  have  the  diameter  of  the 
smallest  circle. 

"Reasoning  from  this  value  the  diameters  of  the  other  circles 
are  obtained." 


Then  x*  +  lox  -f-  25  =  A, 

22  (3  x*  +  iox  -f-  25)  =  B, 

and   7  •  4  •  120  +  B  =  C  =  22y2,  where  ir  =  —  . 

That   the   formula   for   C  is    correct   is    seen   by    substituting   for    120   the 
difference  in  the  areas  as  stated.     We  then  have 

7  •  4 


22    fy*        (*  +  5)z       2x»\ 

•  -    <  ---  ----  >  -\-  £>  =  C, 

7    14  4  4  / 


or  22  (yz  —  x*  —  10  x  —  25  —  2  x*  +  3^2  -f~  10*  -+•  25)  =  C, 

or  22_y2  =  C,  which  is,  as  stated  in   the  rule,   "the  product  of  the  square  of 
the  diameter  of  the  largest  circle  multiplied  by  the  moment  of  circumference." 

i  I.e.,  22  x  (x*  +  10^  +  25)  —  D. 

*  I.  e.,  {C  [4  (x  +  5)  -  *]  -  D}*  =5  X. 

3  I.  e.,  22  •  22^*  (x  -\-  5)2  [4  (x  -j-  5)  -f-  2  x]*.      This  is  merely  the   second 
part  of  the  preceding  paragraph  stated  differently. 

4  I.  e.,    X  =  222  (3  xy*   4.  5  ;j/2  —  x*)*,      and     this     quantity    equals 
22  *  y2  (x  -|-  5)2  (6x  -j-  2o)2.     Their  difference  is  a  sextic. 

5  As  explained  on  page  53. 

7 


98  VI.  Seki  Kowa. 

It  may  add  to  an  appreciation  or  an  understanding  of  the 
mathematics  of  this  period  if  we  add  Takebe's  analysis. 

Let  x  be  the  diameter  of  the  largest  circle,  y  that  of  the 
middle  circle,  and  z  that  of  the  smallest  circles.1 


Then    let    AC=  a,    AD  =  b  ,    AB  «  c,    and    BC=d,   these 
being  auxiliary  unknowns  at  the  present  time. 
Then 

2  a  =  —  z  4-  x, 
and 

4  a2  =  z2  —  2  zx  +  x* 
or 

4  a2  —  s2  =  —  2  zx  +  A'2. 
Therefore 

X\  (l) 


i  Takebe  of  course  expresses  these  quantities  in  Chinese  characters.  The 
coefficients  are  represented  by  him  in  the  usual  sangi  form,  where  \x,  -\.y 
and  \\xy  stand  respectively  for  x,  —  y,  and  2xy.  This  notation  is  called 
the  bosho  or  side-notation  and  is  mentioned  later  in  this  work.  Expressions 
containing  an  unknown  are  arranged  vertically,  and  other  polynomials  are 
arranged  horizontally.  Thus  for  x,  —  a-\-x,  a2  —  2  ax  -j-  x*  we  have 

O  I"  I     «2 

I         I       *• 

I 

respectively,  while  for  a2  -|-  2  ab  -}-  b*  we  have 

\a2       \\a(>        \b* 
with  Chinese  characters  in  place  of  these  letters. 


VI.   Seki  Kowa.  99 

If  we  take  y  from  x  we  have  — y  +  x,  which  is  2c. 

Squaring. 

4  c*  =y2  —  2  yx  +  x*.  (2) 

To  y  add  2  and  we  have 

2d=y  +  z. 
Squaring, 

4  d2  =y2  +  2yz  +  z2. 

Subtracting  z*S  we  have 

4  (^  +  0s  =72  +  27*. 

Subtract  from  this  (i)  and  (2)  and  we  have 
&xSc=  2yz  +  (22  +  2y)  x—2x*. 
Dividing  by  2, 

bx$c=yz+  (z  -\r y)  x  —  x2.  \ 

Squaring, 

bzx  i6c*=y*z2  +  (2y*z  +  2yz*)  x  +  (y  +  z)2  x2 

—  (2y  +  2z)  x*  +  x*.  (3) 

Multiplying  (i)  by  (2)  we  also  have 
b2x.  \6c2  —  —  2y2zx  +  (y2  +  4yz)x2  —  (2y  -f  2z}x*  +  x*, 
which  being  canceled  with  the  expression  in  (3)  gives 

y*z2  +  (4y*z  +  2yza)x  +  (—  4yz  +  z2)x=  o, 
from  which,  by  canceling  z, 

y2z  +  (4y*  +  2yz)x  +  (— 47  +  z)x2  =  o. 
This  may  be  written  in  the  form 

y2z  +  (x2z  —  4x2y)  +  (4y2  +  2yz}x  =  o. 

Takebe  has  now  eliminated  his  auxiliary  unknowns,  and  he 
directs  that  the  quantity  in  the  first  parenthesis  be  squared 
and  canceled  with  the  square  of  the  rest  of  the  expression, a 

1  And  noting  that  d* — (-1-)  z*  =  (b-\-cy. 

2  This  amounts  to  equating  x*z  —  4*2y   to  — [j/2z  +  (4^*  -j-  zvz)  x],  and 
then  squaring  and  canceling  out  like  terms. 

7* 


IOO 


VI.  Seki  Kowa. 


and  that  the  rest  of  the  steps  be  followed  as  in  Seki's  solution. 
In  this  he  expresses  y  and  z  in  terms  of  x  and  given  quantities 
and  thus  finds  an  equation  of  the  sixth  degree  in  x.  Without 
attempting  to  carry  out  his  suggestions,  enough  has  been  given 
to  show  his  ingenuity  in  elimination. 

The  1 2th  problem  proposed  by  Sawaguchi  is  as  follows: 
There   is   a    triangle   in  which   three  lines,   a,  b,  and  c,  are 
drawn  as  shown  in  the  figure.     It  is  given  that 

a  =  4,  b  =  6,  c  =  1.447, 

that  the  sum  of  the  cubes  of  the  greatest  and  smallest  sides 
is  637,  and  that  the  sum  of  the  cubes  of  the  other  side  and 
of  the  greatest  side  is  855.  Required  to  find  the  lengths  of 
the  sides. 


Seki  solves  this  problem  by  the  use  of  an  equation  of  the 
54th  degree. 

The  1 4th  problem  is  of  somewhat  the  same  character.  It 
is  as  follows: 

There  is  a  quadrilateral  whose  sides  and  diagonals  are  re- 
presented by  u,  v,  w,  x,  y,  and  z,  as  shown  in  the  figure. 


W 


VI.  Seki  Kowa.  IOI 

It  is  given  that 

S3 U3  =  271 

U3 2/3=217 

^3 y3=     6O.8 

j3  —  TJU  3  =  326.2 

w^ —  x*  •=    61. 

Required  to  find  the  values  of  u,  v,  w,  x,  y,  and  z.* 

Seki  does  not  state  the  equation  that  is  to  be  solved,  but 
he  says: 

"To  find  z  we  have  to  solve  by  the  reversed  method  an 
equation  of  the  145 8th  degree.  But  since  the  analysis  is  very 
complicated  and  cannot  be  stated  in  a  simple  manner  we  omit 
it,  merely  hinting  at  the  solution. 

"Take  the  'celestial  element'  for  z,  from  which  the  expressions 
of  the  cubes  of  u,  v,  w,  x,  and  y  may  be  derived. 

"Then  eliminate  x*,  the  analysis  leading  to  an  equation  of 
the  1 8th  degree. 

"Next  eliminate  ?f3,  leading  to  an  equation  of  the  54th  degree. 

"Next  eliminate  y ',  leading  to  an  equation  of  the  162  d  degree. 

"Next  eliminate  v^,  leading  to  an  equation  of  the  486th  degree. 

"Now  by  eliminating  u3  two  equal  expressions  result  from 
which  the  final  equation  of  the  145  8th  degree  is  obtained. 
Solving  this  equation  by  the  reversed  method  we  obtain  the 
value  of  z.  This  method2  of  analysis  leads  us  to  the  result 
step  by  step  and  may  serve  as  an  example  of  the  method  of 
attacking  difficult  problems." 

Seki's  explanation  is,  as  he  states,  very  obscure.  Undoubt- 
edly he  explained  the  work  orally  to  his  pupils,  with  the  sangi 
at  hand.  As  the  matter  stands  in  his  statement  it  would  appear 
that  he  had  five  equations  with  six  unknowns  and  that  he  had 


i  This  is  exactly  as  in  the  original,  except  that  symbols  replace  the  words. 
With  merely  these  equations  it  is  indeterminate.  Takebe  adds  another 
equation,  z*  +  «« —  x^  =  z,zs,  where  s  is  the  projection  of  u  upon  z. 

3  Essentially  the  method  of  constructing  the  equation. 


IO2  VI.  Seki  Kowa. 

not  made  use  of  the  geometric  relations  involved,  so  that  we 
are  left  to  conjecture  what  particular  equations  he  may  have 
employed. 

Although  the  explanations  given  by  Seki,  as  shown  in  the 
few  examples  quoted,  are  manifestly  incomplete  and  obscure, 
they  are  nevertheless  noteworthy  as  marking  a  step  in  mathe- 
matical analysis.  His  predecessors  had  been  content  to  state 
mere  rules  for  attaining  their  results,  as  were  also  many  of 
the  early  European  algebraists.  Leonardo  of  Pisa,  for  example, 
solves  a  numerical  cubic  equation  to  a  remarkable  degree  of 
approximation,  but  we  have  not  the  slightest  idea  of  his  method. 
Even  in  the  sixteenth  century  the  Italian  and  German  algebraists 
were  content  to  use  the  Latin  expression  "Fac  ita".1  Seki,  how- 
ever, paid  special  attention  to  the  analysis  of  his  problems,  and  to 
this  his  great  success  as  a  teacher  was  largely  due.  His  method 
of  procedure  was  known  as  the  yendan  jutsu,  yendan  meaning  ex- 
planation or  expositon,  and  jutsu  meaning  process,2  a  method 
in  which  the  explanation  was  carried  along  with  the  manipulating 
of  the  sangi  in  the  "Celestial  Element"  calculation  of  the  Chi- 
nese. When  a  problem  arises  in  which  two  or  more  unknowns 
appear  there  are,  in  general,  two  or  more  expressions  involving 
these  unknowns.  These  expressions  Seki  was  wont  to  write 
upon  paper,  and  then  to  simplify  the  relations  between  them 
until  he  reached  an  equation  that  was  as  elementary  in  form 
as  possible.  This  was  in  opposition  to  the  earlier  plan  of 
stating  the  equation  at  once  without  any  intimation  of  the 
method  by  which  it  was  derived.  Moreover  it  led  the  pupil 
to  consider  at  every  step  the  process  of  simplifying  the  work, 
thus  reducing  as  far  as  possible  the  degree  of  the  equation 
which  was  finally  to  be  solved.  ^  Seki's  pupil,  Takebe,  speaks 
enthusiastically  of  his  master's  clearness  of  analysis,  in  these 

1  In  early  German,  thu  ihm  also. 

2  We  might  translate  the  expression  by  the  single  word  analysis. 

3  ENDO  calls   attention   to   the   fact    that  the  yendan  jutsu  may  be  looked 
upon    as   the   repeated   application   of  the   tengen  jutsu   mentioned  on  p.  48. 
See  his  Biography  of  Seki  (in  Japanese)  in  the  Toyo  Gaku-gei  Zasshi,  vol.  14, 
P-  3I3- 


VI.  Seki  Kowa.  103 

words:1  "In  fact  this  yendan  is  a  process  that  was  never  set 
forth  in  China  with  the  same  clearness  as  in  Japan.  It  is  one 
of  the  brilliant  products  of  my  master's  school  and  it  must 
be  agreed  that  it  surpasses  all  other  mathematical  achieve- 
ments, ancient  or  modern." 

These  words  seem  to  be  those  of  an  enthusiastic  disciple 
rather  than  a  simple  chronicler  of  fact,  since  from  the  evidence 
that  is  before  us  the  yendan  was  merely  a  common-sense  form 
of  analysis  such  as  any  mathematician  or  teacher  might  employ, 
although  we  must  admit  that  his  predecessors  had  not  made 
any  use  of  it. 

Takebe  is  not  content,  however,  to  let  Seki's  fame  as  a 
teacher  rest  here,  and  so  he  hints  at  another  and  rather 
esoteric  theory,  as  one  of  the  initiates  of  the  Pythagorean 
brotherhood  might  have  given  mysterious  reference  to  some 
carefully  concealed  principle  of  the  great  master. 

"Although",  he  says,  "there  is  yet  another  divine  method  that 
is  more  far-reaching,  still  I  shall  not  attempt  to  explain  it,  for 
fear  that  one  whose  knowledge  is  so  limited  as  mine  would 
tend  to  misrepresent  its  significance," — a  tribute,  probably, 
to  the  tenzan  method,  Seki's  improvement  upon  that  of  the 
"Celestial  Element".  *  Takebe's  reticence  in  speaking  of  it  may 
merely  have  reflected  the  modesty  of  Seki  himself,  for  of  this 
modesty  we  are  well  assured  by  divers  writers.  To  boast  of 
such  an  invention  would  have  been  entirely  foreign  to  the 
samurai  spirit  of  Seki  and  to  the  exalted  principles  of  Bushido. 
On  the  other  hand,  this  custom  of  secrecy  had  existed  every- 
where before  Seki's  time,  as  witness  the  attitude  of  Tartaglia 
and  Cardan,  and  even  of  a  man  like  Galileo.  In  Japan,  Mori 
is  said  to  have  kept  a  secret  book  that  was  revealed  only  to 
his  most  deserving  pupils,3  and  Isomura  also  had  one,  his 

1  TAKEBE,  Hatsubi  Sampo  Yendan  Genkai,   1685,  preface. 

2  Tenzan  has  a  broader  meaning  that  may  here  be  understood.    It  includes 
practically  all  of  Japanese  mathematics  except  possibly  yenri.    In  a  restricted 
sense   it  is  written   mathematics,    but    it    sometimes  includes   the   "Celestial 
Element"  method. 

3  See  the  Samva  Znihitsu. 


IO4  VI.   Seki  Kowa. 

book  treating  of  the  calculations  relating  to  a  circle  and  an 
arc.1  Seki  was  so  impressed  with  his  discovery  that  he  re- 
vealed it  to  his  most  promising  followers  only  upon  their 
swearing,  with  their  own  blood,  never  to  make  it  public.  And 
so,  for  more  than  half  a  century  after  Seki's  death  the  secret 
remained,  not  becoming  known  to  the  world  until  Arima  Raidd, 
feudal  lord2  of  Kurume,  in  the  island  of  Kyushu,  revealed  it 
in  his  Shuki  Sampo*  in  1769. 

This  method  was  called  by  Seki  the  kigen  seiho,  meaning  a 
method  for  revealing  the  true  and  buried  origin  of  things.  The 
term  suggests  the  title  of  the  papyrus  of  Ahmes,  written  in 
Egypt  more  than  three  thousand  years  earlier,  "The  science 
of  dark  things."  It  would  be  interesting  to  know  the  origin 
and  history  of  this  name  for  algebra  or  certain  algebraic  pro- 
cesses, since  it  is  found  in  various  parts  of  the  world  and  in 
various  ages.  The  tenzan  method  being  the  one  to  which 
Takebe  seems  to  have  referred  in  his  work  of  1685,  we  are 
quite  certain  that  it  was  invented  some  time  before  this  date.4 
It  is  first  called  by  this  name  by  Matsunaga  Ryohitsu.  It 
is  related  that  Lord  Naito  of  Nobeoka,  in  Kyushu,  himself  no 
mean  mathematician,  was  the  one  who  caused  the  adoption 
of  the  name,  requiring  Matsunaga,  a  pupil  of  Araki  who  was 
a  direct  disciple  of  Seki,  to  write  the  Hard-  Yosan  in  which  it 
appears,  s 

The  word  tenzan  consists  of  two  Chinese  ideograms,  ten 
meaning  to  restore,  and  zan  meaning  to  strike  off.  It  would 
be  most  interesting  if  we  could  know  the  relation  (if  any) 
between  this  term  and  the  name  given  by  Mohammed  ibn 
Musa  al-Khowarazmi  (c.  830)  to  his  algebra,—  al-jebr  w'al- 
muqabala,  which  words  mean  substantially  the  same  thing,— 

1  Ibid. 

2  DaimyO. 

3  It  was  in  this  book  that  the  value  of  IT  to  fifty  decimal  places  was  first 
printed  in  Japan,  an  approximation  already  reached  by  Matsunaga. 

4  ENDO,  in  the  Toyo  Gaku-gei  Zasshi,  vol.  14,  p.  314. 

5  OZAWA'S  Lineage  of  Mathematicians  (Japanese),  1801.    The  Horo-Yosan  is 
a  manuscript  without  date. 


VI.   Seki  Kowa.  IO5 

restoration  and  reduction.1  Does  this  resemblance  tell  of  the 
passing  of  the  mystery  of  "the  science  of  dark  things"  from 
one  school  to  another  in  the  perpetual  interchange  of  thought 
in  the  world's  great  republic  of  scholars,  or  are  these  re- 
semblances that  are  continually  met  in  the  history  of  mathe- 
matics mere  coincidences?  This  tenzan  method  may,  however, 
justly  be  called  a  purely  Japanese  product,  the  product  of  Seki's 
brain,  and  quite  unrelated  to  any  Chinese  treatment.2 

We  shall  now  speak  of  the  notation  employed  in  this  method. 
This  notation  is  the  bosho  shiki  already  mentioned.  In  earlier 
times  it  had  been  the  habit  of  Japanese  mathematicians  to  re- 
present numbers  by  the  sangi  method  described  in  Chapter  IV 
and  known  as  the  chu-shiki.*  Seki  amplifies  this  by  writing 
the  numerals  at  the  side  of  a  vertical  line,  the  significance  of 
which  will  be  explained  in  a  moment.  Since  these  numerals 
were  written  at  the  side  of  a  line  this  method  of  writing  them 
is  known  as  bosho  shiki  or  "side  notation".  In  our  explanation 
we  necessarly  use  Latin  letters  and  Hindu-Arabic  forms  instead 
of  the  Chinese  ideograms,  but  otherwise  the  representations 

are  substantially  correct.   Seki  writes  -,  — ,   and  -  -  as  follows: 

3      «  mn 

3)2,  n\  or  «|  i,  mn\abc,  the  numerators  being  placed  on  the 
right  and  the  denominators  on  the  left.  Sometimes  the  vertical 
line  is  replaced  by  sangi  coefficients,  as  in  the  case  of 

r||ir,  27=\\\\\abc,  for  4  ab,    ^,  and^- 
Powers  of  quantities  are  represented  thus: 


la  6 


715 


372  T^  k& 

for   #4,  3tf6^8,  -    „       .     It  will  be  seen  that  the  exponent  in 
each  case  is  one  less  than  that  used  in  occidental  mathematics. 

1  The  varied  fortunes  of  the  name   for  algebra,  in  Europe,  is  interesting. 
Thus  we  have  such  titles  as  algiebr,  algobra,  nmkabel,  almucable,  arte  maggiore, 
ars  magna,  coss,  cossic  art,  and  so  on. 

2  ENDO,  Book  II,  p.  8. 

3  Sangi  notation. 


106  VI.  Seki  Kowa. 

The  reason  is  that  in  the  wasan  as  in  Chinese  mathematics 
the  nth  power  of  a  quantity  is  called  the  "(n — I)  times  self- 
multiplied".  That  is,  the  native  oriental  exponent  shows  not 
the  number  of  factors  but  the  number  of  times  a  quantity  is 
multiplied  by  itself.  The  fractional  exponent  was  not  used  in 
the  native  algebra  of  Japan. 

The  "side  notation"  was  also  used  in  other  ways.    Thus  a  +  b 
might  be  indicated  in  either  of  the  ways  here  shown. 

\l  or  \a  \b 

To  indicate  subtraction  an  oblique  cancelation  line  was  used. 
Thus  b — a  was  indicated  in  these  four  ways: 


It  will  be  noticed  that  this  tensan  notation  was  employed  in 
Seki's  yendan  method.  Indeed  the  tenzan  may  be  considered 
as  the  notation,  while  the  yendan  refers  to  the  method  of  anal- 
ysis. It  is  difficult  to  justify  the  extravagant  praise  of  the 
disciples  of  Seki  with  respect  to  either  of  these  phases  of  his 
work.  He  must  have  been  very  clear  in  his  own  analysis  with 
his  pupils,  and  this  gave  them  a  higher  appreciation  of  the 
yendan  than  anything  that  has  come  down  to  us  would  warrant. 
And  as  for  the  notation,  while  this  is  an  improvement  upon 
that  of  the  Chinese,  the  improvement  does  not  seem  to  have 
been  so  great  as  to  warrant  the  praise  which  it  has  provoked. 
It  was  applied  to  the  entire  range  of  Japanese  mathematics 
except  the  yenri  or  circle  principle,1  but  we  know  that  the 
Chinese  notation  would  have  been  quite  sufficient  for  the  work 
t'o  be  accomplished.  In  its  application  to  factoring,  the  finding 
of  highest  common  factor  and  the  lowest  common  multiple, 
the  summation  of  infinite  series  and  of  power  series  of  the  type 
I*  +  2"  +  3*  +  ...,  the  shosa-ho  or  method  of  differences,  the 
theory  of  numbers,  the  tetsu-jutsu  or  expansion  in  series  of 
the  root  of  a  quadratic  equation,  the  calculation  relating  to 

1  See  ARIMA'S  Shiiki  Sampo,  1769;  ENDO,  Book  II,  pp.  4,  5,  and  in  the 
Toyo  Gaku-gei  Zasski,  vol.  14,  pp.  362 — 364. 


VI.  Seki  Kowa.  IO/ 

regular  polygons,  and  the  study  of  maxima  and  minima,  the 
tensan  notation  seems  to  have  served  its  purposes  fairly  well, 
better  indeed  than  any  notation  known  in  Japan  up  to  that 
time.  How  much  of  this  application  to  the  various  branches 
of  algebra  was  due  to  Seki  and  how  much  to  his  disciples, 
we  shall  never  know.  The  old  Pythagorean  idea  of  ipse  dixit 
seems  to  have  prevailed  in  Seki's  school,  and  the  master  may 
often  have  received  credit  for  what  the  pupil  did. 

Thus  far,  indeed,  we  have  not  found  much  in  the  way  of 
discovery  to  justify  the  high  standing  of  Seki.  It  is  therefore 
well  to  consider  some  of  the  more  serious  contributions  attri- 
buted to  him.  For  this  purpose  we  shall  go  to  a  work  published 
by  Otaka  Yusho  in  1712,  although  compiled  before  1709,  that 
is,  soon  after  Seki's  death.  Otaka  was  a  pupil  of  Araki  Son- 
yei,  who  had  learned  from  Seki  himself,  and  the  book  claims 
to  be  a  posthumous  publication  of  the  works  of  this  master, 
edited  by  Otaka  under  Araki's  guidance.  Although  this  work, 
known  as  the  Katsuyo  Samps  S  does  not  contain  the  tenzan 
system,  it  gives  a  good  idea  of  some  of  Seki's  other  work,  and 
on  this  account  the  publication  was  a  subject  of  deep  regret 
to  the  brotherhood  of  his  followers.  Tradition  says  that  it 
was  owing  to  the  protests  of  these  followers  that  no  further 
publication  of  Seki's  works  was  undertaken  at  a  time  when  an 
abundance  of  material  was  at  hand. 

One   of  the   subjects   treated  in  the   Katsuyo  Sampo  is  the 
shosa-ho  or  shosa  method,  a  theory  that  seems  to  have  arisen 
from  the  study  of  problems  like  the  summation  of  I*  +  2H  +  3* 
+  . . .  Suppose,  for  example,  we  have  such  a  function  as 
P=a1_x  +  a2x2  +  ...  +  anx", 

where  the  coefficients  are  as  yet  undetermined.  Then  if  a 
sufficient  number  of  values  Pt-  are  known  for  various  values  of 
x,  the  various  values  ai  can  be  determined,  and  this  is  one  of 
the  problems  of  the  shosa-ho.  Professor  Hayashi  speaks  of 
the  method  in  general  as  that  of  finite  differences,  and  this 
certainly  is  one  of  its  distinguishing  features. 

1  "A  summary  of  arithmetical  rules." 


108  VI.   Seki  Kowa. 

This  skosa-ho  in  its  general  form  is  not  an  invention  of  Seki's. 
It  appears  to  be  of  Chinese  origin,  perhaps  invented  by  Kuo 
Shou-ching,  a  celebrated  astronomer  of  the  court  of  the  Mogul 
Empire  of  the  I3th  and  I4th  centuries,  and  possibly  even  of 
earlier  origin.  There  are  three  special  forms,  however:  (i)  the 
ruisai  shosa  of  which  an  illustration  has  just  been  given;  (2)  the 
hotel  shosa,  and  (3)  the  konton  shosa,  these  latter  two  being 
first  described  in  the  Shuki  Sampo  of  1 769.  Seki's  contribution 
was,  therefore,  a  worthy  generalization  of  an  older  Chinese 
device,  and  the  application  of  this  improvement  to  new  problems. 

The  shosa-ho  was  doubtless  employed  by  Otaka  in  his 
Katsuyo  Sampo  (1712),  in  which  there  appears  a  table  that 
expresses  the  formulas  for  the  power  series 

Sr  =  ir+2r+y+  ...  +  nr, 

for  r=  i,  2,  3,  ....  N.  Such  power  series  were  called  by 
the  name  hoda,  and  some  of  the  results  of  their  summation 
are  as  follows: 


S2  =~ 


54  =  -L  (6  #s  -j-  1  5  «4 

55  =  -^  (2  n6  +  6n$  +  5  »4  _  ««), 

1  — 

and  so  on  to 

£„  =  —  (2«I2+  i2«"  +  22«10— 

In  Book  III  of  this  same  work,  the  Katsuyo  Sampo,  there  is 
his  Kakuho  narabini  Yendan-Zu,  a  treatment  of  the  subject  of 
regular  polygons,  namely  of  those  of  sides  numbering  3,  4,  ...  20. 
To  illustrate  some  of  the  results  we  shall  consider  the  case 
of  the  apothem  of  a  regular  polygon  of  thirteen  sides. 


VI.  Seki  Kowa.  109 

Using  the  annexed  figure,  as   given  in  the  Katsuyo  Sampo 
(see   Fig.   28    for  the    original),    and    letting   the    side    of  the 


1 

polygon  be  unity,  the  apothem  x,  and  the  radius  y,  we  have 

Now 

(i  +  4-f2)3  =  i  +  I2x2  +  48^  +  0>4x6  —  4.og6xabcde, 

a  statement  made  without  any  explanation.  Otaka  now  pro- 
ceeds by  a  series  of  unproved  statements  to  develop  two 
equations,  viz., 

-  i  +  $\2x2  —  1 14,400 x*  +  109,824^  —  329,472  X*  +  292,864 x10 
-  53,248  x"  =  o, 

from  which  we  are  to  find  x,  the  apothem,  and 


from  which  we  are  to  find  y,  the  radius. 

The  treatment  of  the  circle  is  given  in  Book  IV  of  the 
Katsuyo  Sampo  and  is  similar  to  that  attempted  by  Muramatsu 
in  his  Sanso  of  1663.  A  circle  of  unit  diameter  is  taken,  a 
square  is  inscribed,  and  the  sides  of  the  inscribed  regular  polygon 
are  continually  doubled  until  a  polygon  of  21?  sides  is  reached. 


VI.  SekiKowa. 


o 


X 


Fig.  28.     From  Otaka's  Katsuyo  Sampo  (1712). 


VI.  Seki  Kowa.  Ill 

The  treatment  thus  far  is  not  at  all  original,  but  the  work  is 
carried  farther  than  in  Muramatsu's  treatise  and  it  represents 
about  the  same  state  of  mathematical  progress  that  was  found 
in  Europe  some  fifty  years  earlier  than  Muramatsu,  or  about  a 
century  before  the  death  of  Seki.  Two  new  features,  however, 
appear.  Of  these  the  first  is  that  if  the  perimeters  of  the  last 
three  polygons  are 

«=3-  14159     26487     76985     6708  — 
£=3.   HI59     26523     86591     3571  + 

c=-$.  14159     26532     88992     7759- 
then 

TT  =  b  +  77 —    —      , 
=  3.   14159265359- 

which  reminds  us  of  some  of  the  incorrect  assumptions  of 
the  Antiphon-Bryson  period,  and  of  the  close  of  the  sixteenth 
century  in  Europe. 

The  second  feature  is,  however,  the  interesting  one.    Starting 

with  the  fraction  — ,  if  we  increase  the  denominator  succes- 
sively by  unity,  and  then  increase  the  numerator  successively 
by  4  or  by  3  according  as  the  previous  fraction  is  less  or 
greater  than  the  known  decimal  value  of  rr,  we  shall  obtain  a 
series  of  values  as  follows: 

(1)  Y  =  3,  "Old  value,"  less  than  TT 

(2)  ---  =  3.5,  greater  than  TT 


(4)  7  =  3-25, 
(5)7-3.2, 

(6)  T|  =  3-166  ..., 

(7)  22  =  3. 142857  .. .,       "Exact  value," 


112  VI.  Seki  Kowa. 

(8)  y  =  3.125,  "Chih's  value,"  less  than  rr 

(20)  —  =  3.15,  "Tung  Ling's  value,"  greater  than  TT 

(25)  ~  =  3- 1 6,  "Old  Japanese  value,"  „           „     „ 

(45)  ^  =  3  •  1 5 5  .  •  • ,  "Liu  Chi's  value,"  „          „    „ 

(50)  i^=  3. 14,  "Hui's  (Liu  Hui's)  value,"    less  than  TT 

(113)  ff|=  3.14159292  .  .  .,  greater  than  TT 

The   names    above  quoted    are    given    by  Otaka,    and   are 

"2.  C  C 

probably  those  used  by  Seki.  The  last  value,  — ,  is  not  as- 
signed a  name,  which  seems  to  show  that  Seki  was  not  aware 
of  Tsu  Ch'ung-chih's  measurement  of  the  circle  as  set  forth 
in  his  Chui-sku,  and  recorded  in  Wei  Chih's  Sui-S/tu.1  The 
value  itself  first  appears  in  printed  form  in  Japan  in  the  works 
of  Ikeda  Shoi  (1672),  Matsuda  Seisoku  (1680)  and  Takebe 
Kenko  (1683). 

The  problem  of  computing  the  length  of  a  circular  arc  also 
appears  in  the  Katsuyo  Sampo,  the  formula  being  given  as 
1276900  (d—h}$  a2  =  5 107600^  //  —  23835413  d$  k* 

+  43470240  d*>  h*  —  37997429  d*  fa 
+  1 5047062  d2  /i$  —  1 501025  dJP 
—  281290/27, 

where  d  —  diameter,  h  =  height  of  segment,  and  a  =  length 
of  arc.  In  the  special  case  where  d=  10  and  // =  2  this 
reduces  to 

41841459200  a2  ==  3597849073280. 

The  method2  of  deriving  this  formula  seems  to  have  been 
purely  inductive,  the  result  of  repeated  measurements,  since  the 
explanation  is  so  obscure  as  to  be  entirely  unintelligible. 

1  "Records   of   the   Sui   Dynasty."     This    fact  was   known,    however,   to 
Takebe,   who   mentions   it   in  his  Ftikyu  Tetsujntsu  of  1722.     It  is  also  given 
in  Matsunaga's  Hoyen  Sankyo  of  1739.     See  also  p.  14,  above.     The  original 
Chui-shu  of  Tsu  Ch'ung-chih  has  been  lost. 

2  Perhaps  relates  to  the  shosa  method  in  a  modified  form. 


VI.  Seki  Kowa.  113 

The  volume  of  the  sphere  is  computed  in  the  Katsuyo  Sampd 
(and  also  in  Seki's  Ritsuyen-ritsii-Kai}  in  an  ingenious  manner. 
The  sphere  is  cut  into  50,  100,  and  200  segments  of  equal 
altitude,  the  diameter  being  taken  as  10.  From  this  Otaka  obtains 
in  some  way  the  three  parameters  666.4,  666.6,  666.65,  each 
of  which  he  multiplies  by  —  to  obtain  the  three  volumes.  Calling 
the  parameters  a,  b,  and  c,  he  now  takes  a  mean  in  this  manner: 


as  in  the  case  of  the  circle.     Multiplying  by 

—  =  — — — ,  we  have 
4        4x113 


--- ^'-SXIPQO 

339        678 


•2  r  r 

for   the    required   volume.      This    amounts    to  taking    g^|    for 
^-,  which  means  that  the  formula  v=—  nr*  is  correctly  used. 

One  of  Seki's  favorite  studies  was  the  theory  of  equations, 
a  subject  treated  in  his  works  on  the  Kaiho  Hempen*  the 
Byodai  Meichi,*  the  Daijutsu  Bengi*,  the  Kaiho  Sanshiki*  and 
the  Kaihd  Hengi-jutsuf  In  the  first  of  these  works  he  class- 
ifies equations  into  four  kinds,  the  jensho  shiki  (perfect  equa- 
tions), hcnsho  shiki  (varied  equations),  kosho  shiki  (mixed  equa- 
tions), and  the  musho  shiki  (rootless  equations),  a  system  not 
unlike  those  found  in  the  works  of  the  Persian  and  Arabian 
writers,  the  classification  according  to  degree  being  relatively 
modern  even  in  Europe.  By  a  perfect  equation  he  means 
one  that  has  only  a  single  root,  positive  or  negative.  A  varied 
equation  is  one  in  which  several  roots  occur,  but  all  of  the 
same  sign.  A  mixed  equation  is  one  in  which  several  roots 

1  "Various  topics  about  equations." 

2  Literally,  "On  making  pathological  problems  perfect." 

3  Literally,  "Discussion  on  the  data  of  problems." 

4  Literally,  "Considerations  on  the  solution  of  equations." 

5  Literally,  "On  new  methods  for  the  solution  of  equations." 


114 


VI.  Seki  Kowa. 


occur,  but  not  all  of  the  same  sign.  A  rootless  equation  is 
one  having  neither  a  positive  nor  a  negative  root,  restricted 
as  Seki  was  aware  to  equations  of  even  degree.1 

In  the  Kailio  Hompen2  Seki  treats  of  positive  and  negative 
roots,  and  sets  forth  a  method  called  the  tekizin-hd*  represent- 
ed by  the  following  table: 


o  degree 

i 

i 

i 

I 

i 

i 

i 

ist.     „ 

i 

2 

3 

4 

5 

6 

2d.        „ 

I 

3 

6 

10 

15 

3d.      „ 

i 

4 

10 

20 

4th.    „ 

i 

5 

15 

5th.    „ 

i 

6 

6th.    „ 

I 

The  method  of  deriving  this  table,  analogous  to  that  for  the 
Pascal  Triangle,  is  evident.  Indeed,  the  vertical  columns  are 
simply  the  horizontal  ones  of  the  usual  triangular  array.  Seki 
does  not  tell  how  the  numbers  are  obtained,  and  no  explanation 
seems  to  have  been  given  by  any  Japanese  until  Wada  Nei 
gave  one  in  the  first  half  of  the  nineteenth  century.*  Such  an 
array  is  rather  obvious  and  was  known  long  before  Pascal  or 
even  Apianus  (1527)  published  it.s  Seki  might  have  used  it, 
as  others  in  the  West  had  done,  for  binomial  coefficients,  but 
it  was  not  meant  by  him  for  this  purpose. 

In  his  Dyddai  Meichi  Seki   calls   attention  to  the  fact  that 

*  I.  e.,  in  general.  Of  course  we  have  also  x  =  V — 2,  x  =  iti,  etc.,  as 
well  as  jr3  =  V — 2,  etc.,  although  Seki  makes  no  mention  of  such  forms, 
having  apparently  no  conception  of  the  imaginary  root. 

2  The  Kaiho-Houpen  of  Hayashi's  History,  part  I,  p.  52. 

3  Literally,  "Vanishing  method,"  relating  to  maxima  and  minima. 

4  In  connection  with  his  theory  of  maxima  and  minima. 

5  SMITH,  D.  E.,  Rara  Arithmetica,  Boston,  1908,  p.  155. 


VI.   Seki  Kowa.  115 

the  mensuration  of  the  circle  or  of  any  regular  polygon  requires 
but  a  single  given  quantity;  that  of  a  rectangle  or  pyramid, 
two  given  quantities;  and  that  of  a  trapezoid,  three.  He  then 
designates  as  tendai  (insufficient  problems)  those  problems  in 
which  there  are  not  enough  data  for  a  solution,  while  those 
having  too  many  data  are  designated  as  handai  (excessive 
problems).  He  also  states  that  in  certain  problems,  although 
the  data  are  correct  as  to  number,  no  perfect  answer  is  to  be 
expected,  and  these  problems  he  calls  kyodai  (imaginary).  They 
arise,  he  says,  in  three  cases:  (i)  where  there  is  no  root, 
(2)  where  all  roots  are  negative,  and  (3)  where  the  roots  of 
the  equation  do  not  satisfy  the  conditions  of  the  original  problem. 
To  illustrate  the  latter  case  he  uses  a  simple  problem  involving 
the  elementary  principle  of  geometric  continuity.  He  proposes 
to  find  the  greater  base  of  a  trapezoid  of  altitude  9,  the 
difference  between  the  bases  being  4,  and  the  smaller  base 
being  10  less  than  the  altitude.  The  problem  is  trivial,  the 
smaller  base  being  9 — 10  or  — i,  and  the  greater  being  4 — I 
or  3.  The  smaller  base,  — i,  does  not  appear  to  Seki  to 
satisfy  a  geometric  problem,  so  he  proceeds  with  considerable 
circumlocution  to  explain  what  is  perfectly  obvious,  that  the 
trapezoid  is  a  cross  quadrilateral.  The  question  of  possible 
roots  of  an  equation  is  discussed  at  some  length  but  in  a 
very  elementary  manner. 

Problems  leading  to  equations  with  two  or  more  roots,  or 
with  negative  roots,  or  with  positive  roots  that  do  not  satisfy 
the  conditions  of  the  problems,  are  called  by  Seki  hendai  or 
pathological  problems,  and  were  intended  to  be  transformed 
into  the  ordinary  determinate  cases  by  a  change  in  the  wording. 

In  his  solution  of  numerical  equations  Seki  not  only  used 
the  "celestial  element"  plan  by  which  the  Chinese  had  anti- 
cipated Horner's  Method  as  early  as  1247,  but  he  effected  at 
least  one  improvement  on  the  Chinese  plan,1  unconsciously 
following  a  line  laid  down  by  Newton. 

1  This  is  seen  in  two  manuscript  works  entitled  Kalho  Sanshiki  and 
Kaiho  Hengi-jutsu. 


VI.  Seki  Ko\va. 
For  example,  in  the  equation 


the  "celestial  element"  method  gives  the  first  two  figures  of 
one  root  as  —  1.7.  Proceeding  as  usual  in  Horner's  Method 
we  have  an  equation  of  the  form 


0.29+ 


;r2  =  o. 


Seki  now  takes  —^-  =  0.063,   but  unlike  his  predecessors   he 

treats  this  as  negative  since  the  two  coefficients  are  positive, 
and  proceeds  as  before,  his  next  equation  being  of  the  form 

0.004169  +  4.474  x  +  x2  =  o. 

Repeating  the  process  we  have  for  the  continuation  of  the 
root  —  0.0009318.  Continuing  the  same  process  Seki  obtains 
for  the  root  —  1.76393202250020. 

One  of  Seki's  Seven  Books1  is  devoted  to  magic  squares 
and  circles,  a  subject  to  which  he  may  have  been  led  by  his 
study  (in  1661)  of  a  Chinese  work  by  Yang  Hui.  He  con- 
siders separately  the  magic  squares  with  an  odd  number  and 
an  even  number  of  cells,  and  with  him  begins  the  first  scientific, 
general  treatment  of  the  subject  in  Japan.  He  begins  by  putting 
into  obscure  verse  his  rule  for  arranging  a  square  of  3*  cells. 
It  would  have  been  impossible  to  make  out  the  meaning  had 
Seki  not  given  the  square  in  a  subsequent  part  of  his  manu- 


script.    As  here   shown   the    square   is  the  common  one  that 
was  well  known  long  before  Seki's  time.     Upon  his   method 


1  The  Hojin  Yensan,  (Hojin  Ensan)  revised  in  manuscript  in  1683.  Araki 
gave  to  these  the  name  of  "Seven  Books"  (Shichibusho),  and  these  he  handed 
down  to  his  disciples. 


VI.  Seki  Kowa. 


117 


for  a  square  of  32  cells  he  bases  his  general  rule  for  one  of 
(2n-f  i)2  cells,  and  this  is  substantially  as  follows: 

Begin  with  the  cell  next  to  the  left  of  the  upper  right-hand 
corner    and   number    to   the   right   and   down   the    right-hand 


12 

ii 

10 

5 

4 

i 

2 

47 

3 

44 

6 

43 

7 

42 

8 

4i 

9 

48 

39 

40 

45 

46 

49 

38 

column  until  n  is  reached.  In  the  annexed  figure  we  have  a 
square  of 

(2n  +  i)2  =  (2.3  +  i)2  =  72  cells. 

We  therefore  number  until  3  is  reached.  Then  go  to  the  left, 
from  the  cell  to  the  left  of  i,  until  2n  —  I  (in  this  case 
2.3  —  i  =  5)  is  reached.  Then  continue  down  the  right  side 
to  the  cell  preceding  the  lower  right-hand  one,  giving  6,  7,  8,  9. 
Then  continue  along  the  top  row  until  the  upper  left-hand 
corner  is  reached,  giving  10,  n,  12.  This  leaves  the  left-hand 
column  to  be  completed,  and  the  lower  row  to  be  filled.  This 
is  done  by  filling  all  except  the  corner  cells  by  the  comple- 
ments to  (2n  +  i)2  +  i  of  the  respective  numbers  on  the  oppo- 
site side,  --in  this  case  the  complements  to  the  number  50. 
Thus,  50  —  3  =  47,  50  —  6  =  44,  and  so  on.  The  corner 
cells  are  complements  to  50  of  the  opposite  corners. 

The  next  step  is  to  take  n  figures  to  the  left  of  the  upper 
right-hand  corner  and  interchange  them  with  the  corre- 
sponding ones  in  the  lower  row,  and  similarly  for  the  n  figures 


n8 


VI.  Seki  Kowa. 


above  the  lower  right  hand  corner.     The  square  then  appears 
as  here  shown. 


12 

ii 

10 

45 

46 

49 

2 

47 

3 

44 

6 

7 

43 

8 

42 

9 

4i 

48 

39 

40 

5 

4 

i 

38 

To  fill  the  inner  cells  Seki  follows  a  similar  rule,  except 
that  the  numbers  now  begin  with  13.  Without  entering  upon 
the  exact  details  it  will  be  easy  for  the  reader  to  trace  the 
plan  by  studying  the  result  as  here  shown.  The  innermost 
square  of  32  cells  is  filled  by  the  method  first  given. 


12 

ii 

10 

45 

46 

49 

2 

47 

20 

19 

35 

37 

14 

3 

44 

34 

24 

29 

22 

16 

6 

7 

17 

23 

25 

27 

33 

43 

8 

18 

28 

21 

26 

32 

42 

9 

36 

3i 

15 

13 

30 

4i 

48 

39 

40 

5 

4 

i 

38 

The  even-celled  squares  have  always  proved  more  trouble- 
some than  the  odd-celled  ones.  Seki  first  gives  a  rule  for  a 
square  of  42  cells,  with  the  result  as  here  shown.  He  then 


VI.  Seki  Kowa. 


divides  these  squares    into   those    that   are    simply   even   and 
those  that  are  doubly  even.1 


4 

9 

5 

16 

1.4 

7 

ii 

2 

15 

6 

10 

3 

I 

12 

8 

13 

For  the  simply  even  squares  above  4*,  Seki  begins,  with  the 
third  cell  to  the  left  of  the  upper  right-hand  corner,  preceding 
thence  to  the  left,  as  shown  in  the  figure.  Then  he  goes  back 
to  the  upper  right-hand  cell  (for  5,  in  the  case  here  shown) 
and  proceeds  down  the  right-hand  column  to  the  third  cell 
from  the  bottom.  He  then  fills  the  vacant  cell  at  the  top 


4 

3 

2 

i 

9 

5 

3i 

6 

30 

7 

29 

8 

27 

10 

32 

34 

35 

36 

28 

33 

(in  this  case  with  9),  and  puts  the  next  number  (10)  in  the 
next  cell  in  the  right-hand  column.  The  remaining  cells  in 
the  left-hand  column  and  the  lower  row  are  complements 
of  the  corresponding  numbers  with  respect  to  4  (n  +  i)2  +  I, 
there  being  2  (n  +  i)  elements  on  a  side,  as  in  the  case  of  an 
odd-celled  square.  The  interchange  of  elements  is  now  made 
in  a  manner  somewhat  like  that  of  the  odd-celled  square, 


1  [2  («  +  i)]2,  and  [2  (2 n)]2. 


120 


VI.  Seki  Kowa. 


the  result   being   here   shown  for  the  case  of  a  square  of  62 
cells.     The  rest  of  the  process  is  as  in  the  odd- celled  case. 


4 

3 

35 

36 

28 

5 

6 

3i 

30 

7 

8 

29 

10 

27 

32 

34 

2 

I 

9 

33 

For  the  doubly  even  magic  square  the  first  step  of  Seki's 
method  will  be  sufficiently  understood  by  reference  to  the 
following  figure,  in  which  the  number  is  82.  The  inner  squares 
are  filled  in  order  until  the  one  of  42  cells  is  reached,  when 
that  is  filled  in  the  manner  first  shown. 


6 

5 

4 

3 

2 

I 

8 

7 

56 

9 

55 

10 

54 

ii 

53 

12 

52 

13 

5i 

14 

58 

60 

61 

62 

63 

64 

57 

59 

Seki  simplified  the  treatment  of  magic  circles,  giving  in  sub- 
stance the  following  rule: 

Let  the  number  of  diameters  be  n.  Begin  with  i  at  the 
center  and  write  the  numbers  in  order  on  any  radius,  and  so 


VI.  Seki  Kowa. 


121 


on  along  the  next  n — i.  Then  take  the  radius  opposite  the 
last  one  and  set  the  numbers  down  in  order,  beginning  at  the 
outside,  and  so  on  along  the  rest  of  the  radii.  In  Fig.  29  the 
sum  on  any  circle  is  140,  and  for  readers  who  have  not  be- 
come familiar  with  the  Chinese  numerals  the  following  diagram, 
although  arranged  for  only  thirty  three  numbers,  will  be  of  service: 


In  another  of  Seki's  manuscripts T  there  appears  the  Josephus 
problem  already  mentioned  in  connection  with  Muramatsu. 

Mention  should  be  made  of  Seki's  work  on  the  mensuration 
of  solids,  which  appears  in  two  of  his  manuscripts.2  He  begins 


1  Sandaisu  Kempu  (Kenpti). 

2  The    Kyitseki   (Calculation    of    Areas    and   Volumes)    and    the    Kyuketsu 
ciigyo  So   (An  incomplete  treatise  on  the  volume  of  a  sphere). 


122 


VI.  Seki  Kowa. 


by  considering  the  volume  of  a  ring1  generated  by  the  revolu- 
tion of  a  segment  of  a  circle  about  a  diameter  parallel  to  the 
chord  of  the  segment.  He  states  that  the  volume  is  equal  to 


Fig.  29.     Magic  circle,  from  the  Seki  reprint  of  1908. 

the  product   of  the    cube    of  the    chord    and  the  moment  of 
spherical  volume.* 

He  finds  this  volume  by  taking  from  the  sphere  the  central 


1  He  calls  it  an  "arc-ring,"  kokan  or  kokwan  in  Japanese. 

2  That  is,  the  volume  of  a  unit  sphere.    It  is  called  by  Seki  the  ritsu-yen 
seki  ritsu  or  gyoku  seki  ho. 


VI.  Seki  Kowa. 


123 


cylinder    and  the  two  caps.1     He    also   considers    the  case  in 
which  the  axis  cuts  the  segment. 


B 


-0 


He  likewise  finds  the  volume  generated  by  a  lune  formed 
by  two  arcs,  the  axis  being  parallel  to  the  common  chord, 
and  either  cutting  the  lune  or  lying  wholly  outside.  Such 
work  does  not  seem  very  difficult  at  present,  but  in  Seki's 
time  it  was  an  advance  over  anything  known  in  Japan.2  These 
problems  were  to  Japan  what  those  of  Cavalieri  were  to  Europe, 
making  a  way  for  the  Katsujutsu  or  method  of  multiple  inte- 
gration ^  of  a  later  period. 

Seki  also  concerned  himself  with  indeterminate  equations, 
beginning  with  ax  —  by  =  I,  to  be  solved  for  integers.4  His 
first  indeterminate  problem  is  as  follows:  "There  is  a  certain 
number  of  things  of  which  it  is  only  known  that  this  number 
divided  by  5  leaves  a  remainder  I,  and  divided  by  7  leaves  a 
remainder  2.  Required  the  number." 


1  This   is   stated   by   an   anonymous    commentary   known   as   the   Kyiiketsu 
Hertgyo  So  Genkai. 

2  ENDO,  Book  II,  p.  45. 

3  Or  rather  the  method  of  repeated  application  of  the  tetsujittsu  expansion. 
Some  of  the  problems  involved  only  a  single  integration. 

4  This  appears  in  his  Shiii  Shoyaku  no  Ho,  written  in  1683.     His  method 
of   attacking    these    problems    he   calls    the   senkan  futsu.     Problems   of   this 
nature  appeared  in  the  Kivatsuyo  Sampo. 


124  VI-   Seki 

Since   the   number    is   evidently   $x  +  I,    and    also    'jy  +  2, 

we  have 

Sx+  i  =  77  +  2, 

whence  5  y— 7y  =  i , 

which  is  in  the  form  that  he  is  considering.  By  what  he 
calls  the  "method  of  leaving  unity",  he  solves  and  finds  that 
#=3,  jp  =  2,  and  the  number  is  16.  He  then  proceeds  to 
generalize  the  case  for  any  number  of  divisors.1 
Seki  also  gives  the  following  typical  problem: 
"There  is  a  certain  number  of  things  of  which  it  is  only 
known  that  this  number  multiplied  by  35  and  divided  by  42 
leaves  a  remainder  35;  and  multiplied  by  44  and  divided  by 
32  leaves  a  remainder  28;  and  multiplied  by  45  and  divided 
by  50  leaves  a  remainder  35.  Required  the  number."  His 
result  is  13  and  it  is  obtained  by  a  plan  analogous  to  the 
one  used  in  the  first  problem.  His  other  indeterminate  problems 
show  a  good  deal  of  ingenuity  in  arranging  the  conditions, 
but  it  is  not  necessary  to  enter  further  into  this  field. 

One  of  the  most  marked  proofs  of  Seki's  genius  is  seen  in 
his  anticipation  of  the  notion  of  determinants.2  The  school  of 
Seki  offered  in  succession  five  diplomas,  representing  various 
degrees  of  efficiency.  The  diploma  of  the  third  class  was 
called  the  Fukudai-menkyo,  and  represented  eighteen  or  nineteen 
subjects.  The  last  of  these  subjects  related  to  the  fukudai 
problems  or  problems  involving  determinants,  and  since  it 
appears  in  a  revision  of  i683,3  its  discovery  antedates  this 
year.  Leibnitz  (1646 — 1716),  to  whom  the  Western  world 
generally  assigns  the  first  idea  of  determinants4,  simply  asserted 

1  Jo-ichi  jutsu.     He  seems  to  have   taken  it   from  the  Chinese  method  of 
Ch'in  Chiu-shao  as  set  forth  in  the  Su-shu  Chiu-chang  of  1247. 

2  T.  HAYASHI,  The   "Fukudai"   and  Determinants  in  Japanese  Mathematics. 
Tokyd  Sugaku-Buturigakkwai  Kizi,  vol.  V  (2),  p.  254  (1910). 

3  The  Fukudai-wo-kaisuru-ho  or  Kai-fukudai-no-ho  (Method  of  solving  fukudai 
problems). 

4  T.  MuiR,   Theory  of  Determinants  in    the  historic  order  of  its  development. 
London,   1890;    D.   E.    SMITH,    History   of  Modern   Mathematics.     New   York, 
1906,  p.  26. 


VI.  Seki  Kowa,  12$ 

that  in  order  that  the  equations 

IO+  \\X-\-  I2J  =  O,      20+ 21*+ 22J=O,     30  +  3  I  #  +  32j  =  O 

may  have  the  same  roots  the  expression 

10.21.32  —  10.22.31—11.20.32+  11.22.30  +  12.20.31—12.21.30 

must  vanish.1  On  the  other  hand,  Seki  treats  of  n  equations. 
While  Leibnitz's  discovery  was  made  in  1693  and  was  not 
published  until  after  his  death,  it  is  evident  that  Seki  was  not 
only  the  discoverer  but  that  he  had  a  much  broader  idea  than 
that  of  his  great  German  contemporary.  To  show  the  essential 
features  of  his  method  we  may  first  suppose  that  we  have 
two  equations  of  the  second  degree, 

axz  +  bx  +  c  —  o 
ax*  +  b'x  +  c  =  o. 

Eliminating  x*  we  have 

(a  b  —  ab')  x  +  (a  c  —  ac')  =  o, 

and  eliminating  the  absolute  term  and  suppressing  the  factor  x 
we  have 

(ac — a  c)  x  +  {be  — b' c)  =  o. 

That  is,  we  have  two  equations  of  the  second  degree  and 
transform  them  into  two  equations  of  the  first  degree  by  what 
the  Japanese  called  the  process  of  folding  (tataimi).  In  the 
same  way  we  may  transform  n  equations  of  the  wth  degree 
into  n  equations  of  the  n — I  degree.2  From  these  latter 
equations  the  wasanka*  proceeded  to  eliminate  the  various 
powers  of  x.  Since  it  was  their  custom  to  write  only  the 
coefficients,  including  all  zero  coefficients,  and  not  to  equate 
to  zero,4  their  array  of  coefficients  formed  in  itself  a  deter- 
minant, although  they  did  not  look  upon  it  as  a  special  function 
of  the  coefficients.  On  this  array  Seki  now  proceeds  to  per- 

*  See  MUIR,  loc.  cit.,  p.  5. 

2  Called  Kwanshiki  (substitute  equations). 

3  Follower  of  the  wasan  (native  mathematics). 

4  The  second  member  always  being  zero  in  a  Japanese  equation. 


126  VI.  Seki  Kowa. 

form  two  operations,  the  san  (to  cut)  and  the  chi  (to  manage). 
The  san  consisted  in  the  removal  of  a  constant  literal  factor 
in  any  row  or  column,  exactly  as  we  remove  a  factor  from 
a  determinant  today.  If  the  array  (our  determinant)  equalled 
zero,  this  factor  was  at  once  dropped.  The  chi  was  the  same 
operation  with  respect  to  a  numerical  factor. 

Seki  also  expands  this  array  of  coefficients,  practically  the 
determinant  that  is  the  eliminant  of  the  equations.  In  this 
expansion  some  of  the  products  are  positive  and  these  are 
called  set  (kept  alive),  while  others  are  negative  and  are  called 
koku  (put  to  death),  and  rules  for  determining  these  signs  are 
given.  Seki  knew  that  the  number  of  terms  in  the  expansion 
of  a  determinant  of  the  «th  order  was  n\,  and  he  also  knew 
the  law  of  interchange  of  columns  and  rows.1  Whatever,  there- 
fore, may  be  our  opinion  as  to  Seki's  originality  in  the  yenri,2 
or  even  as  to  his  knowledge  of  that  system  at  all  or  as  to 
its  value,  we  are  compelled  to  recognize  that  to  him  rather 
than  to  Leibnitz  is  due  the  first  step  in  the  theory  which  after- 
wards, chiefly  under  the  influence  of  Cramer  (1750)  and  Cauchy 
(1812),  was  developed  into  the  theory  of  determinants.^  The 
theory  occupied  the  attention  of  members  of  the  Seki  school 
from  time  to  time  as  several  anonymous  manuscripts  assert,4 
but  the  fact  that  nothing  was  printed  leads  to  the  belief 


1  The   details   of   these    laws    as    expressed   by   the   wasanka   of  the'  Seki 
school   have  been   made   out   with  painstaking   care   by   Professor   HAYASHI, 
and  for  them  the  reader  is  referred  to  his  article. 

2  See  Chapter  VIII. 

3  The   best   source   for  the   history   of  the    subject   in   the  West  is  MUIR, 
loc.  cit. 

4  Professor  HAYASHI  has   several  in  his   possession.     An  anonymous  one 
that  seems  to  have  been  written  in  the  eighteenth  century,  entitled  Fukudai 
riu  san  ka  yendan  justsu,  is   in  the  library  of  one    of  the    authors    (D.  E.  S.\ 
A  contemporary  of  Seki's,  Izeki   Chishin,   published   a  work    entitled   Sampo 
Hakki  in  1690,  in  which  the  subject  of  determinants  is  treated,  and  upwards 
of  twenty   other   works   on   the    subject   are   now  known.     It  is   strange  that 
the    Japanese    made    no    practical    use    of  the   idea  in   connection   with   the 
solution  of  linear  equations,  and  entirely  forgot  the  theory  in  the  later  period 
of  the  wasan. 


VI.  Seki  Kowa.  I2/ 

that  the  process  long  remained  a  secret.  It  must  be  said, 
however,  that  the  Chinese  and  Japanese  method  of  writing 
a  set  of  simultaneous  equations  was  such  that  it  is  rather 
remarkable  that  no  predecessor  of  Seki's  discovered  the  idea 
of  the  determinant. 

We  have  now  considered  all  of  Seki's  work  save  only  the 
mysterious  yenri,  or  circle  principle.  It  must  be  confessed 
that  aside  from  his  anticipation  of  determinants  the  result  is 
disappointing.  In  Chapter  VIII  we  shall  consider  the  yenri, 
of  which  there  is  grave  doubt  that  Seki  was  the  author,  and 
aside  from  this  and  his  discovery  of  determinants  his  reputation 
has  no  basis  in  any  great  field  of  mathematics.  That  he  was 
a  wonderful  teacher  there  can  be  no  doubt;  that  he  did  a 
great  deal  to  awaken  Japan  to  realize  her  power  in  learning 
no  one  will  question;  that  he  was  ingenious  in  improving 
mathematical  devices  is  evident  in  everything  he  attempted; 
but  that  he  was  a  great  mathematician,  the  discoverer  pf  any 
epoch-making  theory,  a  genius  of  the  highest  order,  there  is 
not  the  slightest  evidence.  He  may  be  compared  with  Christian 
Wolf  rather  than  Leibnitz,  and  with  Barrow  rather  than  Newton. 
When,  on  November  15,  1907,  His  Majesty  the  Emperor  of 
Japan  paid  great  honor  to  his  memory  by  bestowing  upon 
him  posthumously  the  junior  class  of  the  fourth  Court  rank, 
he  rendered  unprecedented  distinction  to  a  great  scholar  and 
a  great  teacher,  but  not  to  a  great  discoverer  of  mathematical 
theory. 


t   . 


CHAPTER  VII. 
Seki's  contemporaries  and  possible  Western  influences. 

Whether  or  not  Seki  can  be  called  a  great  genius  in  mathe- 
matics, certain  it  is  that  his  contemporaries  looked  upon  him 
as  such,  and  that  he  reacted  upon  them  in  such  way  as  to 
arouse  among  the  scholars  of  his  day  the  highest  degree  of 
enthusiasm.  Although  he  followed  in  the  footsteps  of  Pythagoras 
in  his  relations  with  his  pupils,  admitting  only  a  few  select 
initiates  to  a  knowledge  of  his  discoveries,1  and  although  he 
kept  his  discoveries  from  the  masses  and  gave  no  heed  to  the 
researches  of  his  contemporaries,  nevertheless  the  fact  that  he 
could  accomplish  results,  that  he  could  solve  the  puzzling 
problems  of  the  day,  and  that  he  had  such  a  large  following 
of  disciples,  made  him  a  stimulating  example  to  others  who 
were  not  at  all  in  touch  with  him.  In  view  of  this  fact  it  is 
now  proposed  to  speak  of  some  of  Seki's  contemporaries  before 
considering  his  own  relation  to  the  yenri,  and  at  the  same 
time  to  consider  the  question  of  possible  Western  influence  at 
this  period. 

Two  years  before  Seki  published  (1674)  his  Hatsubi  Sampo; 
namely  in  1672,  Hoshino  Sanenobu  published  his  KokOgen-sho, 
and  in  1674  Murase,  a  pupil  of  Isomura,  wrote  the  Sampo 
Futsudan  Kai.  A  year  later  (1675),  Yuasa  Tokushi,  a  pupil 
of  Muramatsu,  published  in  Japan  the  Chinese  Suan-fa  Tung- 
tsong.  In  1 68 1  Okuda  Yuyeki,  a  Nara  physician,  wrote  the 
Shimpen  Sansu-ki.  Two  years  later,  Takebe  Kenko  published 


1  A  custom   always   followed   in   the   native  Japanese   schools,  not  merely 
in  mathematics  but  also  in  other  lines. 


VII.  Seki's  contemporaries  and  possible  Western  influences.       129 

the  Kenki  Sampo,  in  which  he  solved  the  problems  proposed 
in  Ikeda  Shoi's  Sugaku  Jojo  Orai  of  1672,  without  making  use  of 
the  tenzan  algebra  of  Seki,  saying  that  "this  touches  upon 
what  my  mathematical  master  wishes  kept  secret,"  thus  leaving 
unsolved  those  problems  that  required  the  senkan-jntsu  and 
similar  devices.  It  was  in  the  work  of  Ikeda  that  the  old 

3  C  C 

Chinese  value  of  TT,  -— ,  was  first  made  known  in  Japan. 

In  the  same  year  (1683)  Kozaka  Sadanao  published  his 
Kuichi  Sangaku-sho?  He  had  been  the  pupil  of  a  certain 
Tokuhisa  Komatsu,  founder  of  the  Kuichi  school  of  mathe- 
matics, a  school  that  was  much  given  to  astrology  and 
mysticism.2  Also  in  this  year  Nakanishi  Seiko  published  his 
Kokogen  Tekito-sku,  a  book  that  was  followed  in  1684  by  the 
Sampo  Zoku  Tekito-shu  written  by  his  brother,  Nakanishi  Seiri. 
These  brothers  had  been  pupils  of  Ikeda  Shoi,  and  one  of 
them  3  opened  a  school  called  after  his  name. 

In  1684  the  second  edition  of  Isomura's  Ketsugi-sho  appeared,4 
and  in  the  following  year  Takebe's  commentary  on  Seki's 
Hatsubi  Sampo  was  published.  This  latter  made  generally 
known  the  yendan  method  as  taught  by  Seki. 

In  1687  Mochinaga  and  Ohashi  published  the  Kaisan-ki 
Komokup  and  in  1688  the  Tdsho  Kaisanki.6  In  the  first  of 
these  works  we  already  find  approaches  to  the  crude  methods 
of  integration  (see  Fig.  30)  that  characterized  the  labors  of 
the  early  Seki  school.  In  the  year  1688  Miyagi  Seiko,  the 
teacher  of  Ohashi,  published  the  Meigen  Sampo,  to  be  followed 
in  1695  by  his  Wakan  Sampo  ^  in  which  he  considers  in  detail 
the  numerical  equation  of  the  1458th  degree  already  mentioned 
by  Seki,  and  attempts  to  solve  the  hundred  fifty  problems 


1  Literally,  the  Mathematical  Treatise  of  the  Kuichi  School. 

2  ENDO,  Book  II,  p.  18. 

3  The   eldest,    Nakanishi   Seiko,    may   have   studied    under   one   of  Seki's 
pupils.     ENDO,  Book  II,  p.  20. 

4  See  p.  65. 

5  Literally,  the  Summary  of  Kaisan-ki. 

6  Literally,  the  Kaisan-ki  with  Commentary. 

7  Japanese  and  Chinese  Mathematical  Methods. 

9 


130      VII.  Seki's  contemporaries  and  possible  Western  influences. 


in   Sato's    Kongenki   and    the    fifteen    in    Sawaguchi's    Kokon 
Sampo-ki  (1670),  all  by  the  yendan  process. 

Miyagi  founded  a  school  in  Kyoto  that  bore  his  name,  and 
to  him  is  sometimes  referred  a  manuscript1  on  the  quadrature 
of  the  circle.  He  was  highly  esteemed  as  a  scholar  by  his 
contemporaries.* 

In  1689  Ando  Kichiji  of  Kyoto  published  a  work  entitled 
Ikkyoku  Sampo  in  which  the  yendan  algebra  is  set  forth,  and 


§  fj  §  9  8  8  9  fl  9  fl  #  fl  '3  l3  3\  $ 


Fig.  30.     Early  integration,  from  Mochinaga  and  Ohashi's 
Kaisan-ki  Komoku  (1687). 

in    1691    Nakane   Genkei   published   a   sequel   to   it  under   the 
title  Shicliijo  Beki  Yenshiki. 

In  1696,  Ikeda  Shoi  published    a   pamphlet  on  the  mensur- 
ation  of  the   circle    and    sphere^   and    in    1698   Sato   Moshun 

1  The    Kohal    Shokai.      This    is,    however,    an    anonymous    work    of   the 
eighteenth  century. 

2  ENDO,  Book  II,  p.  29. 

3  The  Gyokuyen  Kyoku-seki,   the  Limiting  Values   of  the  circular  Area  and 
spherical   Volume.     In  the   same    year   (1696)   Nakane  Genkei   published  his 
Tenmon    Zukwai    Hakki,    an    astronomical    work    of    importance.      The    best 
astronomical  treatise  of  this  period   is  Shibukawa  Shunkai's  Tenmon  Keilo,  a 
manuscript   in   8  vols.     Nakane    Genkei    also   wrote  a  work  on  the  calendar, 
the  Kmva  Tsureki  that  was  later  revised  by  Kitai  Oshima. 


VII.  Seki's  contemporaries  and  possible  Western  influences.       131 


Fig.  31.     Mensuration  of  the  circle,  from  Sato  Moshun's 
Tengen  Shinan  (1698). 


9* 


132      VII.   Seki's  contemporaries  and  possible  Western  influences. 

published  his  Tengen  Shinan  or  Treatise  on  the  Celestial 
Element  Method.  In  this  his  method  of  finding  the  area  of  a 
circle  is  distinctly  Western  (Fig.  31),  although  it  is  so  simple 
as  to  claim  no  particular  habitat. 

This  list  is  rather  meaningless  in  itself,  without  further 
description  of  the  works  and  a  statement  of  their  influence 
upon  Japanese  mathematics,  and  hence  it  may  be  thought  to 
be  of  no  value.  It  is  inserted,  however,  for  two  purposes: 
first,  that  it  might  be  seen  that  the  Seki  period,  whether  through 
Seki's  influence  or  not,  whether  through  the  incipient  influx  of 
Western  ideas  or  because  of  a  spontaneous  national  awakening, 
was  a  period  of  special  activity;  and  second,  that  it  might  be 
shown  that  out  of  a  considerable  list  of  contemporary  writers, 
only  those  who  in  some  way  came  under  Seki's  influence 
attained  to  any  great  prominence. 

We  now  turn  to  the  second  and  more  important  question, 
did  Seki  and  his  contemporaries  receive  an  impetus  from  the 
West?  Did  the  Dutch  traders,  who  had  a  monopoly  of  the 
legitimate  intercourse  with  mercantile  Japan,  carry  to  the 
scholars  of  Nagasaki  and  vicinity,  where  the  Dutch  were 
permitted  to  trade,  some  knowledge  of  the  great  advance  in 
mathematics  then  taking  place  in  the  countries  of  Europe  ? 
Did  the  Jesuit  missionaries  in  China,  who  had  followed  Matteo 
Ricci  in  fostering  the  study  of  mathematics  in  Peking,  succeed 
in  transmitting  some  inkling  of  their  knowledge  across  the 
China  Sea?  Or  did  some  adventurous  scholar  from  Japan  risk 
death  at  the  order  of  the  Shogun,1  and  venture  westward  in 
some  trading  ship  bound  homewards  to  the  Netherlands?  These 
are  some  of  the  questions  that  arise,  and  which  there  are 
legitimate  reasons  for  asking,  but  they  are  questions  that  future 
research  will  have  more  definitely  to  answer.  Some  material 
for  a  reply  exists,  however,  and  the  little  knowledge  that  we 
have  may  properly  be  mentioned  as  a  basis  for  future  in- 
vestigation. 

It  has  for  some  time  been  known,  for  instance,  that  there 


1  Even  the  importation  of  foreign  books  was  suppressed  in   1630. 


VII.  Seki's  contemporaries  and  possible  Western  influences.       133 

was  a  Japanese  student  of  mathematics  in  Holland  during 
Seki's  time,1  doubtless  escaping  by  means  of  one  of  the  Dutch 
trading  vessels  from  Nagasaki.  We  know  nothing  of  his 
Japanese  name,  but  the  Latin  form  adopted  by  him  was 
Petrus  Hartsingius,  and  we  know  that  he  studied  under  Van 
Schooten  at  Leyden.  That  he  was  a  scholar  of  some  distinc- 
tion is  seen  in  the  fact  that  Van  Schooten  makes  mention  of  him 
in  his  Tractatns  de  concinnandis  demonstrationibus  geometricis 
ex  calculo  algebraico  in  one  of  his  editions  of  Descartes's  La 
Gcoincfrie,2  as  follows:  "placuit  majoris  certitudinis  ergo 
idem  Theorema  Synthetice  verificare,  procendo  a  concessis 
ad  quaesita,  prout  ad  hoc  me  instigavit  praestantessimus  ac 
undequaque  doctissimus  juvenis  D.  Petrus  Hartsingius,  lapo- 
nensis,  quondam  in  addiscendis  Mathematis,  discipulus  meus 
solertissimus."^  The  passage  in  Van  Schooten  was  first 
noticed  by  Giovanni  Vacca,  who  communicated  it  to  Professor 
Moritz  Cantor. 

Some  further  light  upon  the  matter  is  thrown  by  a  record 
in  the  Album  Studiosontm  Acadcmiae  Lngduno  Batavae,1'  as 
follows: 

"Petrus  Hartsingius  Japonensis,  31,  M.  Hon.  C."  with  the 
date  May  6,  1669.  Here  the  numeral  stands  for  the  age  of 
the  student,  M.  for  medicine,  his  major  subject,  and  Hon.  C. 
for  Honoris  Causa,  his  record  having  been  an  honorable  one. 

1  HARZER,  P.,  Die  exaclen  Wissemchaften  im  alien  Japan,  Jahresbericht  der 
dcittschen  Mathematiker-Vereinigung,   Bd.  14,    1905,    Heft  6;    MIKAMI,  Y.,   Zur 
Frage  abendliindischer  Einfliisse    auf  die  japanische  Mathematik    am   Ende  des 
sicbzehnten  Jahrhunderts,  Bibliotheca  Mathematica,  Bd.  VII  (3),   Heft  4. 

2  HARZER   quotes   from   the    1661  edition,   p.  413.     We  have  quoted  from 
the  Amsterdam  edition  of  1683,  p.  413. 

3  T.  HAYASHI  remarks  that  the  same  words  appear  in  a  posthumous  work 
of  Van  Schooten's,  but  this  probably  refers  to  the  above  editio  tertia  of  1683. 
See  HAYASHI,  T.,  On  the  Japanese  who  was  in  Europe  about  the  middle  of  the 
seventeenth  century  (in  Japanese),  Journal  of  the  7"okyo  Physics  School,  May,  1905; 
MIKAMI,    Y.,  Hatono  Soha  and  the  mathematics   of  Sek'i,   in  the  Nieuw  ArchieJ 
i'oo>-  Wiskitnde,  tweede  Reeks,  Negende  Deel,   1910. 

4  Hague,    1875.     It   gives    a   list    of   students   and   professors   from    1575 
to  1875. 


134       VII.  Seki's  contemporaries  and  possible  Western  influences. 

Mathematics,  his  first  pursuit,  had  therefore  given  place  to 
medicine,  and  in  this  subject,  as  in  the  other,  he  had  done 
noteworthy  work.  Possibly  the  death  of  Van  Schooten  in  1661 
may  have  influenced  this  change,  but  it  is.  more  likely  that 
the  common  union  of  mathematics  and  medicine,  as  indeed 
of  all  the  sciences  in  those  days,1  led  him  to  combine  his  two 
interests.  Moreover  certain  other  records  inform  us  that  Hart- 
singius  lived  in  the  house  of  one  Pieter  van  Nieucasteel  by 
the  Langebrugge,  a  bit  of  information  that  adds  a  touch  01 
reality  to  the  picture.  This  record  would  therefore  lead  to 
the  belief  that  he  was  only  twenty-two  years  old  when  he  was 
mentioned  in  the  year  of  Van  Schooten's  death  (1661),  or 
probably  only  twenty-one  when  he,  a  doctissimus  juvenis,  and 
quondam  in  addiscendis,  verified  the  theorem  for  his  teacher. 
A  careful  examination  of  the  Leyden  records  as  set  forth 
in  the  Album  Studiosorum  throws  a  good  deal  more  light  on 
the  matter  than  has  as  yet  appeared.  In  the  first  place  the 
Hartsingius  was  adopted  as  a  good  Dutch  name,  it  appearing 
in  such  various  forms  as  Hartsing  and  Hartsinck,  and  may 
very  likely  have  belonged  to  the  merchant  under  whose 
auspices  the  unknown  student  went  to  Holland.  In  the  next 
place,  Hartsingius  was  in  Holland  for  a  long  time,  fifteen  years 
at  least,  and  was  off  and  on  studying  in  the  university  at 
Leyden.  He  is  first  entered  on  the  rolls  under  date  August  29, 
1654,  as  "Petrus  Hartsing  Japonensis.  20,  P,"  a  boy  of  twenty 
in  the  faculty  of  philosophy.  This  would  have  placed  his  birth 
in  1634  or  1635,  but  as  we  shall  see,  he  was  not  very  par- 
ticular as  to  exactness  in  giving  his  age.2  He  next  appears 
on  the  rolls  in  the  entry  of  date  August  28,  1660,  "Petrus 
Hartzing  Japonensis,  22,  M."  He  has  now  changed  his  course 
to  medicine,  and  his  age  would  now  place  his  birth  in  1638 
or  1639,  four  years  later  than  stated  before.  Since,  however, 


1  Witness,  for  example,  the   mention  made  by  Van  Schooten  in  the   1683 
edition    (p.   385)     above    cited,    of    the    assistence    received    from    Erasmius 
Bartholinus,  mathematician  and  physician  in  Copenhagen. 

2  See  Album,  col.  438. 


VII,  Seki's  contemporaries  and  possible  Western  influences.       135 

the  difficulty  of  language  is  to  be  considered,  together  with 
the  fact  that  such  records,  hastily  made,  are  apt  to  be  in- 
exact, this  is  easily  understood.  He  next  appears  in  the 
Album  under  date  May  6,  1669,  as  already  sfated.  He  there- 
fore began  in  1654,  and  was  still  at  work  in  1669,  but  he  had 
not  been  there  continuously. 

Further  light  is  thrown  upon  his  career  by  the  fact  that  he 
was  not  alone  in  leaving  Japan,  perhaps  about  1652.  He  had 
with  him  a  companion  of  the  same  age  and  of  similar  tastes. 
In  the  Album,  under  date  September  4,  1654,  appears  this 
entry:  "Franciscus  Carron  Japonensis,  20,  P."  Within  a  week, 
therefore,  of  the  first  enrollment  of  Hartsingius,  another  Japanese 
of  same  age,  and  doubtless  his  companion  in  travel,  registered 
in  the  same  faculty.  But  while  Hartsingius  remained  in  Leyden 
for  years,  we  hear  no  more  of  Carron.  Did  he  die,  leaving 
his  companion  alone  in  this  strange  land?  Did  he  go  to  some 
other  university?  Or  did  he  make  his  way  back  to  Japan?1 

Now  who  was  this  Petrus  Hartsingius  who  not  only  braved 
death  by  leaving  his  country  at  a  time  when  such  an  act  was 
equivalent  to  high  treason,  but  who  was  excellent  as  a  mathe- 
matician? What  ever  became  of  him?  Did  he  die,  an  unknown 
though  promising  student,  in  some  part  of  the  West,  or  did 
he  surreptitiously  find  his  way  back  to  his  native  land?  If  he 
passed  his  days  in  Europe  did  he  send  any  messages  from 
time  to  time  to  his  friends,  telling  them  of  the  great  world  in 
which  he  dwelt,  and  in  particular  of  the  medical  work  and  the 
mathematics  of  the  intellectual  center  of  Northern  Europe?  In 
other  words,  for  our  immediate  purposes,  could  the  mathe- 
matics of  the  West,  or  any  intimation  of  what  was  being 
accomplished  by  its  devotees,  have  reached  Japan  in  Seki's 
time? 


1  SCHOTEL,  G.  D.  ].,  De  Academie  (e  Leiden  in  de  i6e,  i?e  en  i8e  eeuw 
Haarlem,  1875,  speaks  (p.  266)  of  Japanese  students  at  Leyden,  and  a  further' 
search  may  yield  more  information.  We  have  been  over  the  lists  with  much 
care  from  1650  to  1670,  and  less  carefully  for  a  few  years  preceding  and 
following  these  dates. 


136     VII.  Seki's  contemporaries  and  possible  Western  influences. 

These  questions  are  more  easily  asked  than  answered,  but 
it  is  by  no  means  improbable  that  the  answers  will  come  in 
due  time.  We  have  only  recently  had  the  problem  stated, 
and  the  search  for  the  solution  has  little  more  than  just  begun, 
while  among  all  of  the  literature  and  traditions  of  the  Japanese 
people  it  is  not  only  possible  but  probable  that  the  future 
will  reveal  that  for  which  we  are  seeking. 

At  present  there  is  a  single  possible  clue  to  the  solution. 
We  know  that  a  certain  physician  named  Hatono  Soha,  who 
flourished  in  the  second  half  of  the  seventeenth  century,  did 
study  abroad  and  did  return  to  his  native  land.1  Hatono  was 
a  member  of  the  Nakashima2  family,  and  before  he  went  abroad 
he  was  known  as  Nakashima  Chozaburo.  The  family  was  of 
the  samurai  class,  and  formerly  had  been  retainers  of  the 
Lord  of  Choshu  or  of  the  Lord  of  Iwakuni,3  feudal  nobles 
who  had  made  the  Nakashimas  at  one  time  abundantly  wealthy, 
but  who  had  dishonestly  deprived  them  of  much  of  their  means 
during  the  infancy  of  two  of  the  heirs.  It  was  because  of  this 
wrong  that  the  family  had  left  their  former  home  and  service 
and  had  repaired  to  the  island  of  Kyushu  to  seek  to  mend 
their  fortunes.  It  was  thus  that  they  came  to  Nagasaki,  and 
that  the  young  Nakashima  Chozaburo  met  a  Dutch  trader 
with  whom  he  departed  into  the  forbidden  world  beyond  the 
boundaries  of  the  empire.  It  would  seem,  now,  that  we  ought 
to  be  able  to  ascertain  the  date  of  the  departure  of  the  young 

*  For  much  of  this  information  we  are  indebted  to  S.  Hatono,  a  lineal 
descendent  of  the  physician  in  question,  and  bearing  his  name.  He  informs 
us  that  the  story  was  originally  recorded  in  a  manuscript  entitled  Tsuboi  Idan 
which  was  destroyed  by  fire.  See  also  ISHIGAMI,  T.,  Hatono  Soha  0  in  the 
Chiigivai  Iji  Shimpo,  no.  369,  Aug.  5,  1895;  YOKOYAMA,  T.,  A  physician  of 
the  Dutch  school  who  went  abroad  two  centuries  ago,  and  his  surgical  instruments 
(in  Japanese),  in  the  Kyoyuku  Gakujutsu  Kai,  vol.  4,  January  1901,  (an  article 
that  leaves  much  to  be  desired  in  the  matter  of  clearness);  FUJIKAWA,  Y., 
History  of  Japanese  Medicine  (in  Japanese);  YOKOYAMA,  T.,  History  of  Education 
in  Japan  (in  Japanese). 

2  In  the  eastern  part  of  Japan  this  name  commonly  appears  as  Nakajima, 
but  Nakashima  is  the  preferred  form. 

3  The  latter  was  subject  to  the  former. 


VII.  Seki's  contemporaries  and  possible  Western  influences.       137 

samurai,  and  to  trace  his  wanderings,  especially  as  he  returned 
and  could,  at  least  in  the  secrecy  of  his  family,  have  told 
his  story.  We  are,  however,  quite  uncertain  as  to  any  of  these 
matters.  His  descendants  have  kept  the  tradition  that  his  visit 
abroad  was  in  the  Manji  era,  and  since  this  extended  from 
1658  to  1 66 1,  it  included  the  time  that  Hartsingius  was  in 
Leyden.  Tradition  also  says  that  he  visited  the  capital  of 
Namban,  which  at  that  time  meant  not  only  the  Spanish 
peninsula,  but  the  present  and  former  colonies  of  Spain  and 
Portugal,  and  which  included  Holland.  While  in  this  city 
he  learned  medicine  from  someone  whose  name  resembled 
Postow  or  Bostow,1  and  after  some  years  he  again  returned 
to  Japan. 

Arrived  in  his  own  country  Nakashima  was  in  danger  of 
being  beheaded  for  his  violation  of  the  law  against  emigration, 
and  this  may  have  caused  the  journeying  from  place  to  place 
which  tradition  relates  of  him.  It  is  more  probable,  however, 
that  his  skill  as  a  physician  rendered  him  immune,  the  officials 
closing  their  eyes  to  a  violation  of  the  law  which  might  be 
most  helpful  to  themselves  or  their  families  in  case  of  sickness. 
The  danger  seems  to  have  passed  through  the  permission 
granted  by  the  Shogun  that  two  European  physicians,  Almans 
and  Caspar  Schambergen  should  be  permitted  to  practise  at 
Nagasaki.  Thereupon  Nakashima  became  one  of  their  pupils, 
began  to  practise  in  the  same  city,  and  assumed  the  name 
Nakashima  Soha. 

It  happened  that  there  lived  at  that  time  in  the  province 
of  Hizen,  in  Kyushu,  a  certain  daimyo  who  was  very  fond  of 
a  brood  of  pigeons  that  he  owned.  One  of  the  pigeons  having 
injured  its  leg,  the  daimyo  sent  for  the  young  physician,  and 
such  was  the  skill  shown  by  him,  and  so  rapid  was  the  recovery 


i  We  have  been  unable  to  find  this  name  among  the  list  of  prominent 
Spanish,  Portuguse,  or  Dutch  physicians  of  that  time,  but  it  is  not  improbable 
that  some  reader  may  identify  it.  Is  it  possible  lhat  it  refers  to  Adolph 
Vorstius  (Nov.  23,  1597— Oct.  9,  1663)  who  was  on  the  medical  faculty  at 
Leyden  from  1624  to  1663? 


138        VII.  Seki's  contemporaries  and  possible  Western  influences. 

of  the  bird,  that  in  all  that  region  Nakashima's  name  be- 
came known  and  his  praises  were  sung.  So  celebrated  was 
his  simple  exploit  that  the  people  called  him  Hato  no  as  hi  2vo 
naoshita  Sd/ia1  or  Hato  no  Solia?  a  name  so  pleasing  to  him 
that  he  thereupon  adopted  it  and  was  thenceforth  known  as 
Hatono  Soha.3 

His  fame  now  having  found  its  way  along  the  Inland  Sea,  a 
daimyo  of  the  Higo  province,  Lord  Hosokawa,  in  due  time 
called  him  to  enter  his  service  at  Osaka,  so  that  he  left  Naga- 
saki, bearing  with  him  gifts  from  his  masters,  Almans  and 
Schambergen,  as  well  as  those  which  Postow  had  presented 
when  he  was  in  Europe  or  in  some  colony  of  Spain,  Portugal, 
or  Holland.  This  was  in  i6Si,4  and  there  he  seems  to  have 
remained  until  his  death  in  1697,  at  the  a§e  °f  fifty-six  years. 
Such  is  the  brief  story  of  the  only  Japanese  scholar  who  is 
known,  though  native  sources,  to  have  studied  in  Europe  and 
to  have  returned  to  his  own  country  at  about  the  time  that 
Petrus  Hartsingius  was  studying  mathematics  and  medicine  in 
Leyden.  If  Hatono  was  fifty-six  when  he  died,  as  the  family 
records  assert,  he  must  have  been  born  in  1641  which  is  a 
little  too  late  for  Hartsingius,  whereas  if  he  and  Carron  are 
the  same,  his  birth  is  placed  in  1634  or  1635,  which  argues 
strongly  against  this  conjecture. 

The  problem  seems,  therefore,  to  reduce  to  the  search  for 
a  Doctor  Postow,  and  to  a  search  for  some  problem  in  the 
Japanese  mathematics  of  the  Seki  school  that  is  at  the  same 
time  in  Van  Schcoten's  Tractatus  or  in  some  contemporary 
treatise.  Thus  far  we  have  no  knowledge  that  Hatono  knew 


1  Soha  who  cured  the  pigeon's  leg. 

2  Soha  of  the  pigeon. 

3  The  name  is  now  in  the  ninth  generation. 

4  This  is  the  date  as  it  appears   in   the  family  records,   as  communicated 
to  us   by   his   descendant.     According   to  T.  Yokoyama,   however,  there   is  a 
manuscript   in   the  possession  of  the  family,    signed  by  Deshima  Ranshyu  at 
Nagasaki  in  1684.     If  this  is  a  nom  </<?  plume  of  Hatono's   as  Mr.  Yokoyama 
believes,  he  may  have  gone  to  Osaka  later  than  1681. 


VII.  Seki's  contemporaries  and  possible  Western  influences.       139 

any  mathematics  whatever.1  If  he  was  Hartsingius  he  could 
easily  have  communicated  his  knowledge  to  Seki  or  his  dis- 
ciples, and  if  he  was  not  it  is  certain  that  he  would  have 
known  him  if  he  studied  in  Leyden,  and  in  any  case  there  is 
the  mysterious  Franciscus  Carron  to  be  considered. 

As  to  Seki's  contact  with  those  who  could  have  known  the 
foreign  learning,  a  story  has  long  been  told  of  his  pilgrimage 
to  the  ancient  city  of  Nara,  then  as  now  one  of  the  most 
charming  spots  in  all  Japan,  and  still  filled  with  evidence 
of  its  ancient  culture.  It  appears  that  he  had  learned  of 
certain  treatises  kept  in  one  of  the  Buddhist  temples,  that 
had  at  one  time  been  brought  from  China  by  the  priests,2 
which  related  neither  to  religion  nor  to  morals  nor  to  the 
healing  art,  and  which  no  one  was  able  to  understand.  No 
sooner  had  he  opened  the  volumes  than  he  found,  as  he  had 
anticipated,  that  they  were  treatises  on  Chinese  mathematics, 
and  these  he  copied,  taking  the  results  of  his  labor  back  to 
Yedo.  It  is  further  related  that  Seki  spent  three  years  in 
profitable  study  of  these  works,  but  what  the  books  were  or 
what  he  derived  from  them  still  remains  a  mystery.3 

If  Seki  went  to  Nara,  the  great  religious  center  of  Japan, 
as  there  seems  no  reason  to  doubt,  he  would  not  have  failed 
to  visit  the  great  intellectual  center,  Kyoto,  which  is  near  there. 
Neither  would  he  have  missed  Osaka,  also  in  the  same  vicinity, 
where  Hatono  Soha  was  in  the  service  of  the  daimyo.  But 

1  Most  of  his   manuscripts   and   the   records   of  the    family    were   burned 
some  fifty  years  ago,  and  of  the  few  that  remained  nearly  all  were  destroyed 
at  the  siege  of  Kumamoto  at  the  time  of  the  Saigo  rebellion  in  1877. 

2  MIKAMI,  Y.,    On  reading   P.  Harzer1!  paper  on  the  mathematics  in  Japan, 
Jahresbericht  der  deutschen  Math.  Verein.,  Bd.  XV,  p.  256. 

3  Seki  may  have  studied  the  Chinese  work  by  Yang  Hui  at  Nara.    The  story 
of  his  visit  is  said  to  have  first  appeared  in  the  Burin  Inken  Roku  or  Burin 
Kenbun    Roku   written   by   one   SaitO.      It  was   reproduced  in   an   anonymous 
manuscript   entitled   Samoa   Zuihitsu,    possibly  written  by  Furukawa  Ken.     It 
also  appears  in  the  Okinagusa  written  by  Kamizawa  Teikan.    We  have  been 
unable  to  get  any  definite  information  as  to  the  Nara  books,  although  diligent 
inquiry  has  been  made,  but  we  wish  to  express  our  appreciation  of  the  efforts 
in  this  direction  made  by  Mrs.  Kita  (nee  Mayeda)  and  her  brother. 


I4O      VII.  Seki's  contemporaries  and  possible  Western  influences. 

on  the  other  hand,  Seki  published  the  Hatsubi  Sainpo  in  1674, 
while  Hatono  did  not  go  to  Osaka  until  1681,  so  that  in  any 
event  Seki  could  solve  numerical  equations  of  a  high  degree1 
before  Hatono  settled  in  his  new  home.  Moreover  the  symbolism 
used  by  him  is  manifestly  derived  from  the  Chinese,2  so  that 
this  part  of  his  work  shows  no  European  influence.  If  Hatono 
or  Hartsingius  influenced  Seki  it  must  have  been  in  the  work 
in  infinite  series,  which,  as  we  shall  see  in  the  next  chapter, 
started  in  his  school,  although  more  probably  with  his  pupil 
Takebe. 

Still  another  contact  with  the  West  is  mentioned  in  a  work 
called  the  Nagasaki  Scmmin  Den,  in  which  it  is  stated  that 
one  Seki  Sozaburo  learned  astronomy  from  an  old  scholar 
who  had  been  to  Macao  and  Luzon.  If  this  is  the  Luzon  of 
the  Philippine  Islands  he  could  at  that  period  have  come  in 
contact  with  the  Jesuits,  and  this  is  very  likely  the  case. 

Mention  should  also  be  made  of  another  possible  medium 
of  communication  with  the  West  in  the  time  of  Seki.  Aside 
from  the  evident  fact  that  if  Hatono,  Hartsingius,  and  Carron 
ventured  forth  on  a  voyage  to  Europe,  others  whose  names 
are  not  now  remembered  may  have  done  the  same,  we  have 
the  record  of  two  men  who  were  in  touch  with  Western 
mathematics.  These  men  were  Hayashi  Kichizaemon,  and  his 
disciple  Kobayashi  Yoshinobu,  both  of  them  interpreters  in  the 
open  port  of  Nagasaki.  Each  of  these  men  knew  the  Dutch 
language,  and  each  was  interested  in  the  sciences,  the  latter 
being  well  versed  in  the  astronomy  of  the  WesU  Kobayashi 
was  suspected  of  being  a  convert  to  Christianity,  and  as  this 
was  a  period  of  relentless  persecution  of  the  followers  of  this 
religion4  he  was  thrown  into  prison  in  1646,  remaining  there 

1  He  even  hints  at  one  of  the  1458  th  degree     (See  page  129.) 

2  Possibly  obtained  from  Chinese  works  at  Nara. 

3  In  1650  a  Portuguese   whose  Japanese  name  was   Sawano  Chiian  wrote 
an   astronomical  work   in  Japanese,  but  in  Latin   characters.     In  1659  Nishi 
Kichibei  transliterated  it  and  it  was  annotated  by  Mukai  Gensho  (1609  —  1677) 
under  the  title  Kenkon  Benselsu. 

4  It  was  in  1616   that  the   Tokugawa  Shogunate   ordered   the   strict   sup- 


VII.  Seki's  contemporaries  and  possible  Western  influences.       141 

for  twenty-one  years.  Upon  his  release  in  1667  he  made  an 
attempt  to  teach  astronomy  and  the  science  of  the  calendar 
at  Nagasaki,1  though  with  what  success  is  unknown,  and  it  is 
recorded  that  in  the  year  of  his  death,  1683,  at  the  age  of 
eighty-two,  he  was  able  to  correct  an  error  in  the  computation 
of  an  eclipse  of  the  sun  as  recorded  in  the  official  calendar.2 
Hayashi  was  executed  in  1646.  While  it  is  probable  that 
these  men  did  not  know  much  of  the  European  mathematics 
of  the  time,  it  is  inconceivable  that  they  were  unaware  of  the 
general  trend  of  the  science,  and  that  they  should  fail  to  give 
to  inquirers  some  hint  as  to  the  nature  of  this  work. 

A  little  later  than  the  time  of  Kobayashi  there  appeared 
still  another  scholar  who  knew  the  Dutch  astronomy,  one 
Nishikawa  Joken,  who  was  invited  by  the  Shogun  Yoshimune 
to  compile  the  official  calendar.  As  already  stated,  the  latter 
was  himself  a  dilletante  in  astronomy,  and  it  was  due  to  his 
foresight  and  to  that  of  Nakane  Genkei  that  the  ban  upon 
European  books  was  raised  in  1720.  From  this  time  on  the 
astronomy  of  the  West  became  well  known  in  Japan,  and 
scholars  like  Nagakubo  Sekisui,  Mayeno  Ryotaku,  Shizuki 
Tadao,  Asada  Goryu,  and  Takahashi  Shiji  were  thoroughly 
acquainted  with  the  works  of  the  Dutch  writers  upon  the 
subject.3 

The  conclusion  appears  from  present  evidence  to  be  that 
some  knowledge  of  European  mathematics  began  to  find  its 


pression   of  Christianity,   the   result   being   such   a  bloody  persecution  that  a 
rebellion  broke  out  at  Shimabara,  not  far  from  Nagasaki,  in   1637. 

1  ENDO,  Book  II,  p.  76. 

2  ENDO,  Book  II,  p.  18. 

3  Mayeno    is    said    to   have    also    had   a    Dutch    arithmetic    in    1772,    but 
the    title   is   not   known.      ENDO,  Book  III,    p.  7.     On    this   question   of   the 
influence   of  the   Dutch   see    HAYASHJ,    T.,    /fay   have   the  Japanese  used  the 
Dutch  books  imported  from  Holland,  in  the  Nieuw  Archiefvoor  IViskunde,  reeks  2, 
deel  7,   1905,  p.  42;  1906,  p.  39,  and  later,  where  it  appears  that  most  of  the 
Dutch  works  known  in  Japan  are  relatively  late.    On  the  interesting  history 
of  the   Portuguese   writer   known   as  Sawano  Chiian,  see  MIKAMI,  Y.,  in  the 
Nieuw  Archiefvoor  Wiskunde,  reeks  2,  deel   10,  and  the  Annals  Scientificos  da 
Academia  Polytechnica  do  Porto,  vol.  7. 


142       VII.  Seki's  contemporaries  and  possible  Western  influences. 

way  into  Japan  in  the  seventeenth  century;  that  we  have 
no  definite  information  as  to  the  nature  of  this  work  beyond 
the  fact  that  mathematical  astronomy  was  part  of  it;  that  there 
is  no  evidence  that  Seki  or  his  school  borrowed  their  methods 
from  the  West;  but  that  Japanese  mathematicians  of  that  time 
might  very  well  have  known  the  general  trend  of  the  science 
and  the  general  nature  of  the  results  attained  in  European 
countries. 


CHAPTER  VIII. 
The  Yenri  or  Circle  Principle. 

Having  considered  the  contributions  of  Seki  concerning  which 
there  can  be  no  reasonable  doubt,  and  having  touched  upon 
the  question  of  Western  influence,1  we  now  propose  to  examine 
the  yenri  with  which  his  name  is  less  positively  connected. 
The  word  may  be  translated  "circle  principle"  or  "circle  theory", 
the  name  being  derived  from  the  fact  that  the  mensuration  of 
the  circle  is  the  first  subject  that  it  treats.  It  may  have  been 
suggested  by  the  title  of  the  Chinese  work  of  Li  Yeh  (1248), 
the  Tse-yilan  Hai-cJiing,  in  which,  as  we  have  seen  (page  49), 
Tsl-yuan  means  "to  measure  the  circle."  Seki  himself  never 
wrote  upon  it  so  far  as  is  positively  known,  although  tradition 
has  assigned  its  discovery  to  him,  nor  is  it  treated  by  Otaka 
Yusho  in  his  Kivatsuyo  Sampo  of  1712  in  connection  with  the 
analytic  measurement  of  the  circle.  After  Seki's  time  there 
were  numerous  works  treating  of  the  mere  numerical  measure- 
ment of  the  circle,  such  as  the  Taisei  Sankyo,*  commonly 
supposed  to  have  been  written  by  Takebe  Kenko,3  and  of 
which  twenty  books  have  come  down  to  us  out  of  a  possible 
forty-three.4  There  is  a  story,  generally  considered  as  fabulous, 
told  of  three  other  books  besides  the  twenty  that  are  known, 
that  were  in  possession  of  Mogami  Tokunai5  a  century  ago. 


1  The  influence  of  the  missionaries  is  considered  later. 

2  "Complete  Mathematical  Treatise." 

3  So  stated  in  a  manuscript  of  Lord  Arima's  Hoyen  Kiko,  bearing  date  1766. 

4  So  stated   by  Oyamada  Yosei  in  his   article  on   the   Sangaku  Shuban  in 
the  Matsunoya  Hikki,  although  the  number  is  doubtful. 

5  A  pupil  of  Honda  Rimei  (1755—1836). 


144  VI11-  The  Yenri  or  Circle  Principle. 

He  stated  that  he  procured  them  from  one  Shiono  Koteki  of 
Hachioji,  who  had  learned  mathematics  from  Someya  Haru- 
fusa.  Shiono  recorded  these  facts  at  the  end  of  his  copy,  and 
this  is  the  bearing  of  the  story  upon  Seki's  secret  knowledge 
of  the  yenri.  It  was  Someya  who  gave  Shiono  these  books, 
assuring  him  that  they  contained  Seki's  secret  knowledge,  being 
works  that  he  had  himself  written.  Someya  had  received  them 
from  Ishigaya  Shoyeki  of  Kurozawa  in  Sagami,  his  aged  master, 
who  was  a  pupil  of  Seki's  and  who  had  received  these  copies 
from  the  latter's  own  hand. 

Although  the  story  is  not  a  new  one,  and  seems  to  relate 
Seki  intimately  with  the  work,  nevertheless  we  have  no  evidence 
save  tradition  to  corroborate  the  statement,  since  the  three 
volumes  no  longer  exist,  if  they  ever  did,  and  the  twenty  that 
we  know  show  no  evidence  of  being  Seki's  work.1  Moreover 
the  treatment  of  TT  which  it  contains  is  quite  certainly  not  that 
of  Seki,  for  in  his  Fukyu  Tetsujutsu  of  1722  Takebe  states  that 
it  is  not.2  This  treatment  is  based  upon  the  squares  of  the 
perimeters  of  regular  inscribed  polygons  from  4  to  512,  n2 
being  taken  as  the  square  of  the  perimeter  of  the  512-gon, 
namely 

9.86960  44010  89358  61883  449° i  998/4  7- 

Seki,  on  the  contrary,  calculated  the  successive  perimeters 
instead  of  their  squares.  Takebe  claims  to  have  carried  his 
process  far  enough  to  give  TT  to  upwards  of  forty  decimal 
places  by  considering  only  a  iO24-gon,  and  he  gives  it  as 

71  =  3.14159  26535  89793  23843  26433  83279  50288  41971  2.3 

He  then  uses  continued  fractions  to  express  this  value,  stating 
that  this  plan  is  due  to  his  brother  Takebe  Kemmei,  and  that 

1  It  should  be  stated,  however,  that  ENDO  (Book  II,  p.  41)  believes,  and 
with  excellent   reason,   that   they  were    taken    from   Seki's   own  writings  and 
were   put  into   readable  form   by  Takebe.     See   also  MIKAMI,  Y.,  A  Question 
on  Seki's  Invention   of  the  Circle- Principle,    in    the  Tokyo    Sugaku-Buturigakkivai 
A'izi,  Book  IV  (2),  no.  22,  p.  442,  and  also  his  article  on  the  yenri  in  Book  V  (2). 

2  MS.,  article  10. 

3  He  must,  however,  have  gone  beyond  the   TO24-gon  for  this. 


VIII.  The  Yenri  or  Circle  Principle.  145 

Seki  had  used  only  the  method  given  in  the  Kzvatsuyo  Sampo, 
all  of  which  tends  to  throw  doubt  upon  Seki's  connection  with 
this  treatise. 

The  successive  fractions  obtained   for  TT  by  taking  the  con- 
vergents  of  the  continued  fraction  are 


3       22 

333 

355 

i'    7 

'    106  ' 

113' 

103993 
"33102"' 

104348 
33215  ' 

208341 
66317  ' 

312689 
99532 

833719 

i  146408 

4272943 

etc., 

265381  ' 

364913  ' 

1360120  ' 

most  of  which  are  not  found  in  any  work  with  which  we  can 
clearly  connect  Seki's  name. 

Still  another  reason  for  doubting  Seki's  relation  to  this  phase 
of  the  work  is  seen  in  the  method  of  measuring  a  circular  arc. 
In  the  Taisci  Sankyo  the  squares  of  the  arcs  are  used  instead 
of  the  arcs  themselves,  as  in  the  case  of  the  circle.  Some 
idea  of  the  work  of  this  period  may  be  obtained  from  the 
formula  given: 

(4877315687^  +  2 1 309475994*4  A2  +  23945445808^4 
+  5 1 7074 1 462  /i6)a* 

=  4877515687  £8  +  4732289365  3  £6/<:2  +  151469740022^/2* 
+  174277533560^^+  503 19088000  A8, 

where  c  =  chord,  h  =  height  of  arc  (from  the  center  of  the 
chord  to  the  center  of  the  arc),1  and  a  =  length  of  arc.  This 
formula  resembles  one  that  appears  in  the  Kwatsnyo  Sampo, 
and  one  that  is  in  Takebe's  Kenki  Sampo  of  1683.  All  these 
formulas  seem  due  to  Seki. 

Some  idea  of  the  Taisei  Sankyo  having  been  given,  together 
with  some  reasons  for  doubting  the  relation  of  Seki  to  it,  we 
shall  now  speak  of -the  author,  Takebe,  and  of  his  other  works, 
and  of  his  use  of  the  ycnri,  setting  forth  his  testimony  as  to 
any  possible  relation  of  Seki  to  the  method. 


1  Which    we    shall   hereafter   call   the   height   of  the    arc,  the  older  word 
sagilta  being  no  longer  in  common  use. 

10 


146  VIII.  The  Yenri  or  Circle  Principle. 

Takebe  Hikojiro  Kenko1  was  one  of  three  brothers  who 
displayed  a  taste  for  mathematics 2  and  who  studied  under  Seki. 
He  was  descended  from  an  ancient  family,  his  father  Takebe 
Chokuko  being  a  shogunate  samurai.  He  was  born  in  Yedo 
(Tokyo)  in  the  sixth  month  of  1664,  and  while  still  a  youth 
became  a  pupil  of  Seki,  and,  as  it  turned  out,  his  favorite  and 
most  distinguished  one.3 

Takebe  was  only  nineteen  years  of  age  when  he  published 
the  Kenki  Sampo  (1683).  Two  years  later  (1685)  there  ap- 
peared his  commentary  on  Seki's  Hatsubi  Sampo  (of  1674), 
and  in  1690  he  wrote  the  seven  books  of  his  notes  on  the 
Suan-hsiao  Clii-meng  which  appeared  in  his  edition  of  this  work,* 
explaining  the  sangi  method  of  solving  numerical  equations.  In 
1703  he  was  made  a  shogunate  samurai  and  served  as  an 
official  in  the  department  of  ceremonies.  In  1719  he  drew  a 
map  of  Japan,  upon  which  he  had  been  working  for  four  years, 
and  which  for  its  accuracy  and  for  the  delicacy  of  his  work 
was  looked  upon  as  a  remarkable  achievement.  This  and  his 
vast  range  of  scientific  knowledge  served  to  command  the 
admiration  and  respect  of  Yoshimune,  the  eighth  of  the  To- 
kugawa  shoguns,  who  called  upon  him  for  advice  with  respect 
to  the  calendar  and  who  consulted  him  upon  matters  relating 
to  astronomy,  a  subject  in  which  each  took  a  deep  interest. 
He  at  once  pointed  out  certain  errors  in  the  official  calendar, 
and  recommended  as  court  astronomer  Nakane  Genkei,  for 
whom  and  for  himself  Yoshimune  built  an  observatory  in 

1  His    given  name   Kenko    appears    as  Katahiro   in   the   Hakuseki    Shinsho 
written  by   Aral  Hakuseki   (1657  — 1725),    his    contemporary,    and  is  so  given 
in  some    of  the    histories.     It   is  possible   too   that  the  family  name    Takebe 
should    be   Tatebe,    as    given    by    ENDO,   OKAMUTO,    and    others    of  the   old 
Japanese  school,  although  the  former  is  usually  given. 

2  The  other  brothers  were  his  seniors  and  were  called  Kenshi  and  Kemmei, 
also  known  as  Katayuki  and  Kataaki. 

3  KAWAKITA,   C.,  Honcho   Siigaku   Shiryo  (Materials   for   the  Mathematical 
History    of  Japan),  pp.  63  —  66,    this    being   based   upon   Furukawa  Ujikiyo's 
writings.     See  also  Kuichi  Sanjin's  article  in  the  Sugaku  Hochi. 

4  This    Chinese     algebra    appeared    in    1299.       The    Japanese    edition    is 
mentioned  in  Chapter  IV. 


VIII.  The  Yenri  or  Circle  Principle.  147 

the  castle  where  he  dwelt  So  liberal  minded  was  this 
shogun  that  he  removed  the  prohibition  upon  the  importation 
of  foreign  treatises  upon  medicine  and  astronomy,  so  that  from 
this  time  on  the  science  of  the  West  was  no  longer  under 
the  ban. 

The  infirmities  of  age  began  to  tell  upon  Takebe  in  1733 
so  much  as  to  lead  him  to  resign  his  official  position,  and  six 
years  later,  on  the  twentieth  day  of  the  seventh  month  of  the 
year  1739,  he  passed  away  at  the  age  of  seventy-five  years. 

The  work  of  Takebe's  with  which  we  are  chiefly  concerned 
was  written  in  1722,  and  was  entitled  Fukyu  Tetsujutsu,  Fukyu 
being  his  nom  de  plume,  and  Tetsujutsu  being  the  Japanese 
form  of  the  title  of  a  Chinese  work  written  by  Tsu  Ch'ung-chi 
(430 — 501)  in  the  fifth  century.  This  Chinese  work  is  now 
lost,  but  it  treated  of  the  mensuration  of  the  circle,1  and  for 
this  reason  there  is  an  added  interest  in  the  use  of  its  name 
in  a  work  upon  the  yenri. 

Takebe  states2  that  Seki  was  wont  to  say  that  calculations 
relating  to  the  circle  were  so  difficult  that  there  could  be  no 
general  method  of  attack.  Indeed  he  says  that  Seki  was  averse 
to  complicated  theories,  while  he  himself  took  such  delight  in 
minute  analysis  that  he  finally  succeeded  in  his  efforts  at  the 
quadrature  of  the  circle.  It  would  thus  appear  that  the  yenri 
was  not  the  product  of  Seki's  thought,  but  rather  of  Takebe's 
painstaking  labor.  Moreover  the  plan  followed  by  Takebe  in 
finding  the  length  of  an  arc  is  not  the  same  as  the  one  given 
in  the  Kwatsuyo  Sampo  in  which  Otaka  Yusho  (1712)  sets 
forth  Seki's  methods,  though  it  has  some  resemblance  to  that 
given  in  the  Taisei  Sankyo  which,  as  we  have  seen,  Takebe 
may  have  written  in  his  younger  days  when  he  was  more 
under  Seki's  influence. 


1  As  we  know  from  Wei  Chi's  Records  of  the  Sui  Dynasty,  a  work  written 
in  the  seventh  century.     It  was  possibly  a  treatise  on  the  calendar  in  which 
the   circle   was   considered  incidentally.     See  MIKAMI,  Y.,   in   the  Proceedings 
of  the  Tokyo  Math.  Phys,  Society,  October,  1910. 

2  Article  8  of  his  treatise. 

10* 


148  VIII.  The  Yenri  or  Circle  Principle. 

Takebe  takes  a  circle  of  diameter  10  and  finds  the  square 
of  half  an  arc  of  height  o.oooooi  to  be  a  number  expressed 
in  our  decimal  system  as 

o.ooooo  ooooo  33333  35111  11225  39690  66667  28234 
77694  79595  875  + , 

but  he  gives  us  no  complete  explanation  as  to  how  this  was 
obtained.1  Now  since  the  squares  of  the  halves  of  arcs  of 
heights  i,  o.i,  and  o.ooooi,  respectively,  have  for  their  ap- 
proximate values  10,  i,  and  o.oooi,  it  will  be  observed  that 
these  are  the  products  of  the  diameter  and  the  heights  of  the 
arcs.  He  therefore  takes  dh,  the  product  of  the  diameter  and 
height,  as  the  first  approximation  to  the  square  of  half  an  arc. 
He  then  compares  this  approximation  with  the  ascertained 

value  and  takes  his  first  difference  Dr  as  —  h2.  Proceeding  in 
a  similar  manner  he  finds  the  second  difference  D2  to  be 

h          8 

—  •  — -  •  Z>t ,  and  so  on  for  the  successive  differences.  The 
result  is  the  formula 

4  a  3  d     15        T       d  '  14 

h       32       „  h       25       n 

h  ~d'  ~47'      **  *'  33        4 


In  other  words,  he  has 


which  expresses  in   a  series  the  square   of  arc  sin  x  in  terms 
of  versin  x. 

This  series  is  convenient  enough  when  h  is  sufficiently  small, 
but  it  is  difficult  to  use  when  k  is  relatively  large.  Takebe 

1  He  states  that  the  particulars  are  set  forth  in  two  manuscripts,  the 
Yenritsit  (Calculation  of  the  Circle)  and  Koritsu  (Calculation  of  the  Circular 
Arc),  but  these  manuscripts  are  now  lost. 


VIII.  The  Yenri  or  Circle  Principle.  149 

therefore  developed  another  series  to  be  used  in  these  cases, 
as  follows: 

JL  a*  =  dh  +  -  h*  +  ™  .  -*-  •  A  -  -A  •  -L  .  A 

4  3  d—  h       15  </—  >4       14 


</-/4      15  ,/_^     39s 

He  also  gives  a  third  series  which  he,  possibly  following  Seki, 
derives  from  the  value  of  //  =0.00000  oooi,  as  follows: 

!«•-<«  +  -U-+JL--         _.A 

3         IS    d  —  *-h 

H 


, 


_ 
980  6743008 


_ 

26176293  1419 


Takebe's  method  of  finding  the  surface  of  a  sphere  is  the 
same  as  that  given  in  the  revised  edition  of  Isomura's  Ketsugisho 
save  that  it  is  carried  to  a  closer  degree  of  approximation. 
As  bearing  upon  Seki's  work  it  should  be  noted  that  Takebe 
states  that  the  former  disdained  to  follow  this  method,  preferring 
to  consider  the  center  as  the  vertex  of  a  cone  of  which  the 
altitude  equals  the  radius,  showing  again  that  Takebe  was  quite 
independent  of  his  master. 

Not  only  does  Takebe  use  infinite  series  in  the  manner 
already  shown,  but  in  another  of  his  works  he  does  so  in  a 
still  more  interesting  fashion.  This  work  has  come  down  to 
us  in  manuscript  under  the  title  Yenri  Tetsnjntsn  or  Yenri 
Kokai-jiitsu?  In  this  he  considers  the  following  problem:  In 
a  segment  of  a  circle  the  two  chords  of  the  semi-arc  are  drawn, 
after  which  arcs  are  continually  bisected  and  chords  are  drawn. 
The  altitude  of  half  the  given  arc  then  satisfies  the  equation 

-  dh  +  4  dx  —  4  x*  =?  o, 

where  d=  diameter,  h  =  altitude  of  the  given  arc,  x—  altitude 
of  half  of  this  arc.     This  equation  Takebe  proceeds  to  solve 


1  Literally,  The  circle  principle,  or  Method  of  finding  the  arc  of  a  circle. 


150  VIII.  The  Yenri  or  Circle  Principle. 

by  expressing  the  value  of  x  in  the  form  of  a  series,  expanded 
according  to  a  process  which  he  calls  Kijo  Kyftshd  jutsu? 

From  this  expansion  Takebe   derives  a  general  formula  for 
the  square  of  an   arc,  which  he   gives  substantially  as  follows: 

-  -  ,///  22.  4 

~ 


4     ~  3.  4-5-6.  .. 

I 

2"  Hl 


a  result  that  had  previously  been  obtained  in  the  Fukyii 
Tetsujutsu  of  I/22.2 

The  analysis  leading  to  this  formula,  which  is  too  long  to 
be  given  here  and  which  is  obscure  at  best,  is  the  ycnri 
or  Circle  Principle,  and  it  at  once  suggests  two  questions: 
(i)  What  is  its  value?  (2)  Who  was  its  discoverer? 

As  to  each  of  these  questions  the  answer  is  difficult.  In  the  first 
place,  Takebe  does  not  state  with  lucidity  his  train  of  reason- 
ing, and  we  are  unable  to  say  how  he  bridged  certain  diffi- 
culties that  seem  to  have  stood  in  his  way.  He  gives  us  results 
instead  of  a  principle,  an  isolated  formula  instead  of  a  powerful 
method.  To  be  sure  his  formula  has,  as  we  shall  see,  some 
interesting  applications,  as  have  also  many  formulas  of  the 
calculus;  but  here  is  only  one  formula,  obscurely  derived,  whereas 
the  calculus  is  a  theory  from  which  an  indefinite  number  of 
formulas  may  be  derived  by  lucid  reasoning.  We  are  there- 
fore constrained  to  say  that,  from  any  evidence  offered  by 
Takebe,  the  yenri  is  simply  the  interesting,  ingenious,  rather 
obscure  method  of  deriving  a  formula  capable  of  being  applied 
in  several  ways,  but  that  it  is  in  no  more  comparable  to  the 
European  calculus,  even  as  it  existed  in  the  time  of  Seki, 
than  is  Archimedes's  method  of  squaring  the  parabola,  while 
the  method  is  stated  'with  none  of  the  lucidity  of  the  great 
Syracusan. 

1  Literally,  Method  of  deriving  the  root  by  divisions. 

2  See  page  148,  above. 


VIII.  The  Yenri  or  Circle  Principle.  151 

But  taking  it  for  what  it  is  worth,  who  invented  the  yenri? 
The  greatest  of  Japanese  historians  of  mathematics,  Endo,  is 
positive  that  it  was  Seki.  He  sets  forth  the  reasons  for  his 
belief  as  follows:1  "The  inventions  of  the  tenzan  algebra  and 
of  the  yenri  were  made  early  [in  the  renaissance  of  Japanese 
mathematics],  but  certain  scholars  do  not  attribute  the  latter 
to  Seki  for  the  reason  that  it  is  not  mentioned  in  the  Kwatsuyo 
Sampo.  Such  a  view  of  the  question  is,  however,  entirely 
unwarranted.  At  that  period  even  the  tenzan  algebra  was  kept 
a  profound  secret  in  Seki's  school,  never  being  revealed  to  the 
uninitiated.  It  was  on  this  account  that  not  even  the  tenzan 
algebra  was  treated  in  the  Kwatsuyo  Sampo,  and  hence  there 
is  little  cause  for  wonder  that  the  yenri  has  no  place  there. 
It  is  stated,  however,  that  the  value  of  IT  is  slightly  less  than 
3.14159265359.  Now  unless  the  correct  value  were  known 
[to  this  number  of  decimal  places]  how  would  this  fact  have 
been  evident?  .  .  .  The  process  given  in  this  work  being 
restricted  to  the  inscription  of  polygons,  there  was  no  means 
of  knowing  how  many  digits  are  correct.  Nevertheless  the 
author  was  correct  in  his  statement  as  to  how  many  decimal 
places  are  exact,  and  so  it  would  seem  that  he  must  already 
have  known  the  correct  value  to  more  decimal  places  [than 
were  used]  in  order  to  make  his  comparison.  The  original 
source  of  information  was  certainly  one  of  Seki's  writings, 
perhaps  the  same  as  that  used  by  Takebe  in  his  subsequent 
work." 

While  Endo's  argument  thus  far  is  not  conclusive,  since  Seki 
may  have  found  the  value  of  TT  by  the  older  process,  or  may 
have  obtained  it  from  the  West,  nevertheless  it  must  be  granted 
that,  as  Takebe  assures  us,  he  did  know  it  to  more  than  twenty 
figures. 

Endo  continues:  "In  the  Kyoho  era  (1716 — 1736)  Seki's 
adopted  son,  Shinshichi,  was  dismissed  from  office  and  was 
forced  to  live  under  Takebe's  care.  It  was  at  this  juncture 
that  Takebe,  in  consultation  with  him,  entered  upon  a  study 

2  ENDO,  Book  II,  pp.  55,  56. 


152  VIII.  The  Yenri  or  Circle  Principle. 

of  Seki's  most  secret  writing  on  the  yenri  as  applied  to  the 
rectification  of  a  circular  arc,  after  which  he  completed  his 
manuscript  entitled  Yenri  Kohai  Tetsujutsu"  ?  He  continues2  by 
saying  that  Shinshichi  was  dismissed  from  office  in  the  Shogunate 
in  1735  because  of  his  dissolute  character,  so  that  we  thus 
have  a  date  which  will  serve  as  a  limit  for  such  communication 
as  may  have  taken  place.  He  asserts  that  Seki's  adopted  son 
now  gave  to  Takebe  the  secret  writings  of  his  father,  written 
in  the  Genroku  era  (1688  —  1704)  or  earlier,  and  it  was  through 
their  study  that  Takebe  came  to  elaborate  the  yenri.  Endo 
thinks  that  Takebe  did  not  enter  upon  this  work  before  the 
dismissal  of  Seki's  adopted  son  in  1735  at  which  time  he  was 
already  an  old  man.3 

Now  it  .is  evident  that  this  view  of  the  case  is  not  wholly 
correct,  for  Takebe  gives  the  same  series  in  his  Fukyu  Tetsujntsu 
in  1722.  Moreover,  he  must  have  been  acquainted  with  that 
form  of  analysis  because  there  is  extant  a  manuscript  compiled 
in  1728  by  one  Oyama  (or  Awayama)  Shokei4  entitled  Yenri 
Hakki  which  is  quite  like  the  Yenri  Kohai-jutsu  in  its  main 
features,  although  the  work  is  not  so  minutely  carried  out,  in 
spite  of  its  gain  in  simplicity. 

For  example,  the  square  of  the  arc  is  given  in  a  series  which 
is  substantially  the  same  as  the  one  already  assigned  to  Takebe. 
Oyama's  rule  may  be  put  in  modern  form  as  follows: 


[ 


From  this  series  he  derives  the  value  of  TT  by  writing  h=— 


1  ENDO,  Book  II,  p.  74. 

2  Ibid.,  pp.  8 1,  82. 

3  His  reasons  are  not  clear.     Professor  T.  HAYASHI,   in  his  article  in  the 
Honchd  Sugaku  Koenshu,  1908,  pp.  33 — 36,  makes  out  a  strong  case   for  Seki 
as  the  discoverer  of  the  yenri. 

4  Possibly  Tanzan  SkSkei.    The  writer  of  the  preface  of  the  work,  Hachiya 
Teisho,  may  have  been  this  same  person. 


VIII.  The  Yenri  or  Circle  Principle.  153 

and  taking  four   times  the  result.     He  also  finds  it  by  taking 
h  =  d,  the  result  being 


7T'  =  4[] 


Oyama,  the  author  of  the  Yenri  Hakki,  was  a  pupil  of  Kuru 
Juson,  who  had  studied  under  Seki,  but  the  theory  is  not  given 
as  in  any  way  connected  with  the  latter.  In  one  of  the  two 
prefaces  Nakane  Genkei,  a  pupil  of  Takebe's,  says:  "The  most 
difficult  problem  having  to  do  with  numbers  is  the  quadrature 
of  the  circle.  On  this  account  it  is  that  we  have  the  various 
results  of  the  different  mathematicians.  ...  It  is  now  a  century 
since  the  dawn  of  learning  in  our  country,  and  during  this 
period  divers  discoveries  have  been  made.  Of  these  the  most 
remarkable  one  is  that  of  Takebe  of  Yedo.  For  several  decades 
he  has  pursued  his  studies  with  such  zeal  that  oftimes  he  has 
forgotten  his  need  of  food  and  sleep.  In  the  spring  of  1722 
he  was  at  last  rewarded  by  brilliant  success,  for  then  it  was 
that  he  came  upon  the  long-sought  formula  for  the  circle. 
Since  then  he  has  shown  his  result  to  divers  scholars,  all  of 
whom  were  struck  with  amazement,  and  all  of  whom  cried 
out,  'Human  or  divine!  This  drives  away  the  clouds  and 
darkness  and  leaves  only  the  blue  sky!'  And  so  it  may  be 
said  that  he  is  the  one  man  in  a  thousand  years,  the  light 
of  the  Land  of  the  Rising  Sun!" 

The  second  preface  is  by  Hachiya  Kojuro  Teisho,  and  he 
too  gives  the  credit  to  Takebe.  He  says,  "The  circle  principle 
is  a  perfect  method,  never  before  known  in  ancient  or  in 
modern  times.  It  is  a  method  that  is  eternal  and  unchange- 
able ...  It  is  the  true  method,  constructed  first  by  the  genius 
of  Takebe  Kenko,  and  before  him  anticipated  neither  in  Japan 
nor  in  China.  It  is  so  wonderful  that  Takebe  should  have  made 
such  a  valuable  discovery  that  it  is  only  natural  to  look  upon 
him  as  divine.  For  years  have  I  studied  under  Seki's  pupil 
Kuru  Juson,  and  have  labored  long  upon  the  problem  of  the 
quadrature  of  the  circle,  but  only  of  late  have  I  learned  of 


154  VIII.  The  Yenri  or  Circle  Principle. 

Takebe's  discovery,  and  I  shall  be  happy  if  this  work,  which 
I  have  written,  may  initiate  my  fellow  mathematicians  into  the 
mysteries  of  the  problem." 

It  would  seem  from  the  last  sentence  that  Hachiya  may  have 
been  the  real  author  of  the  work,  and  that  Oyama  Shokei  and 
Hachiya  may  have  been  the  same  person.  In  any  case,  however, 
the  evidence  is  clear  that  his  contemporaries  proclaimed  Takebe 
the  discoverer  of  the  yenri,  and  there  seems  to  have  been  none 
to  challenge  this  award.  There  is  no  contemporary  statement 
like  this  that  connects  the  principle  with  Seki,  and  until  there 
is  stronger  evidence  than  mere  conjecture  such  honor  as  is 
due  should  be  bestowed  upon  Takebe. 

But  where  did  Takebe  get  this  formula  for  #2?  His  explan- 
ation of  his  own  development  is  very  obscure.  Did  he  himself 
understand  it,  or  had  he  the  formula  and  did  he  explain  it  as 
far  as  his  ingenuity  allowed:  That  there  is  a  close  resemblance 
between  this  formula  and  such  series  as  one  finds  in  looking 
over  the  works  of  Wallis I  is  evident.  The  series  seems,  however, 
to  have  been  given  by  Pierre  Jartoux,  a  Jesuit  missionary, 
resident  in  Peking.  This  Jartoux  was  born  in  1670  and  went 
to  China  in  1700,  dying  there  Nov.  30,  1720.  He  was  a  man 
of  all-round  intelligence,2  and  his  Observations  astrononiiqucs, 
published  two  years  after  his  death,  showed  some  ability.  He 
also  worked  with  Pere  Regis  on  the  great  map  of  China.  But 
our  interest  in  Jartoux  lies  chiefly  in  the  fact  that  he  was  in 
correspondence  with  Leibnitz,  as  is  shown  by  the  publication 


1  Our  attention   is    called  to   this  fact  by  P.  HARZER,  Die  exakten  Wissen- 
schaften  im  alien  Japan,    in  the  Jahresbericht  der  deutschen  Mathemat.   Vcrcin., 
Bd.  14,  Heft  6.    A  search  through  Wallis  fails,  however,  to  reveal  this  series, 
although  the  analogy  to  this  work  is  evident.     See,  for  example,  WALLIS,  J., 
De  Algebra  Tractatus,  Oxoniae,    1693,   cap.  XCVI.     The    attention  of  readers 
is  invited  to  the  desirability  of  ascertaining  if  this  series  was  already  known 
in  Europe. 

2  His  report,  Details  sur  le  Ging-seng,  et  snr  la  recolte  de  cette  plante,  published 
in  Europe  in  1720,  was  the  best  one  upon  the  subject  that  had  appeared  in 
the  West   up   to   that  time.     Indeed   it  is   for   this  report  that   he   was  best 
known  there. 


VIII.  The  Yenri  or  Circle  Principle.  155 

of  his  Observationes  Macularum  Solarium  Pekino  missae  ad 
G.  W.  Leibnitium  in  the  Acta  Eruditorum? 

Here  then  is  a  scholar,  Jartoux,  in  correspondence  with 
Leibnitz,  giving  a  series  not  difficult  of  deduction  by  the  cal- 
culus, which  series  Takebe  uses  and  which  is  the  essence 
of  the  yenri,  but  which  Takebe  has  difficulty  in  explaining, 
and  which  he  might  easily  have  learned  through  that  inter- 
course of  scholars  that  is  never  entirely  closed.  There  is  a 
tradition  that  Jartoux  gave  nine  series,2  of  which  three  were 
transmitted  to  Japan,  ^  and  it  seems  a  reasonable  conjecture 
that  Western  learning  was  responsible  for  his  work,  that  he 
was  responsible  for  Takebe's  series,  and  that  Takebe  explained 
the  series  as  best  he  could. 

The  knowledge  of  Takebe's  work  was  the  signal  for  the 
appearance  of  various  treatises  upon  the  yenri  besides  that  of 
Oyama,  and  while  they  add  nothing  of  importance  to  the 
theory  or  to  its  history,  mention  should  be  made  of  a  few. 
The  one  that  was  the  most  highly  esteemed  in  the  Seki  school 
of  mathematicians  was  the  Kenkon  no  Maki*  a  work  of  unknown 
authorship. s  Not  only  is  the  author  unknown,  but  the  work 
itself  is  apparently  no  longer  extant  in  its  original  form.6  The 


1  In  1705,  p.  485. 

2  Professor  Hayashi  thinks  that  Jartoux  did  not  give  nine  series,  but  that 
he  gave  six,  and  that   these  were  obtained   by  Ming  An-tu  whose  work  was 
completed   by  his   pupils    after   his   death,   and    published   in    1774.     Among 
these    six    is    Takebe's   series.     Proceedings  of  the    Tokyo  Math.    Phys.    Soc., 
1910  (in  Japanese). 

3  These  three   appear   in   Mei  Ku-cheng's  book,  but   the  date  is  unknown 
and  there  is  no  evidence  that  it  reached  Japan  in  this  period. 

4  Literally,  The  Rolls  of  Heaven  and  Earth. 

5  ENDO  thinks  that  it  was  written  by  Matsunaga;  see  his  History,  Book  II, 
p.  84.     P.  HARZER   thinks   the   author   was  Yatnaji;    see   the  Jahresbericht  der 
deutschen  Morgenl.  Ver.,   Bd.  14,   p.   317.     C.  KAWAKITA   thinks   it  was  Araki, 
and  in  FUKUDA'S  Sampd  Tamatebako  (1879)  the  same  opinion  is  expressed. 

6  A  manuscript  bearing  this  title  was  found  in  a  private  library  at  Sendai, 
in  the  possession  of   a   former  pupil  of  Yamaji,  but  N.  OKAMOTO,  who  has 
investigated  the  matter,  believes   that   it   is  quite    different   from  the  original 
treatise. 


VIII.  The  Yenri  or  Circle  Principle. 

process  followed  in  developing  the  formula  for  a2  is  simpler 
than  that  used  by  Takebe  in  his  Yenri  Kohai-jutsu  and  rather 
resembles  that  of  Oyama. 

The  unknown  author  finds  that  the  altitudes  for  the  successive 
arcs  formed  by  doubling  the  number  of  chords  are 


+  _L  (*.}        *- 

h 


/Ay  , 

W/  J' 


.L,  ,_!_ 

64  7/L          64  W 


4.6      ,/  4.6.8 

s-21  /^\z  5.21.143 

16:40  l7)    +  1640.224 

iiii  /Ay  +  4_i7j7j  /Ay  j 

64.32  W/    +64.32.896  W/  J' 


these  being  calculated  by  the  tetsujutsu  process,  or  the  actual 
expansion  of  the  terms  of  the  equations,  although  the  calcul- 
ations themselves  are  not  given.  The  ratios  of  the  successive 
coefficients  are  seen  to  be 


I-3 

3-5 

5.7         7.9        9.11       n.1.3      13.15 

3-4 

5-6   ' 

7.8    '     9.10  '    11.12'    13  14'    15.16' 

3.5 

7-9 

11.13      15.17      19.21      23.25      27.29 

6.8    ' 

10.  12 

14.16'    18.20'    22.24'    26.28'    30.32' 

7-9 

15.I7 

23-25      31-33      39-41      47.49     55.57 

12.16' 

2O.24 

28.32'    36.40'    44.48      52.56     60.64 

Hence  the  *#th  ratio  for  hr  is  of  the  form 

(km  —  I)  (km  +  1)  2  (kz  m*  —  i) 

~ 


where  k  =  2r,  and  as  k  becomes  infinite  this  reduces  to 

zm* 


We  therefore  have  the  limit  to  which  h  is  approaching,  and 
we  can  compute  the  square  of  the  arc  as  before.  This  is  the 
plan  as  stated  in  the  Sendai  manuscript,  the  only  one  which 
it  seems  safe  to  use,  even  though  the  manuscript  is  evidently 
not  like  the  lost  original.1 

1  ENDO,  Book  II,  pp.  84  —  90,  gives  a  different  treatment,  resembling  that 
found    in  the  Kohai  no  Ri.     None    of  the    leading    mathematicians    of   the 


VIII.  The  Yenri  or  Circle  Principle.  157 

There  is  some  little  testimony  in  favor  of  Seki's  authorship 
of  the  Kenkon  no  Maki,  although  the  presumption  is  entirely 
against  it.  Thus  in  an  anonymous  work  entitled  Kigenkai  or 
Yenri  Kenkon  S/w,  a  note  by  Furukawa  Ujikiyo  relates  the 
following:  "This  book  is  a  writing  of  Seki  Kowa  and  has  long 
been  kept  a  profound  secret.  No  one  into  whose  hands  it  has 
come  was  entitled  to  assume  the  role  of  Seki's  successor.  Hence 
Fujita  Sadasuke  treasured  the  work,  and  copied  it  upon  two 
rolls  which  he  called  Kenkon  no  Maki*  revealing  it  only  to  his 
son  and  to  his  most  celebrated  pupil.  All  this  has  been  told 
me  by  Shiraishi  Chdchu."  The  probabilities  are  that  some 
parts  of  the  work  were  simply  an  ancient  paraphrase  of  Otaka 
Yusho's  Kwatsuyo  Sampo,  and  being  thus  of  the  Seki  school 
it  was  attributed  to  the  master.  Whether  or  not  it  was  the 
original  Kenkon  no  Maki  is  unknown.  However  that  may  be, 
it  extends  the  yenri  to  include  the  analytic  treatment  of  the 
volume  of  a  spherical  segment  of  one  base  of  diameter  a,  by 
a  method  not  unlike  that  of  Cavalieri.  The  segment  is  divided 
into  n  thin  layers  of  diameters  d^,  d2,  ...  dn,  where  d,t=a. 
Then 

72  fj         kh.     kh 

a ,  =  4  (a )  — , 

k  n  '     n    ' 

where  d  =  diameter   of  the    sphere,    and   h  =  altitude   of  the 
segment.     Summing  for  k  =  I,  2,  3,  .  . .  u,  we  have 

y  d*4*.jk-±k-  y*. 

Z-i      k          *    n     jLt  «2      /-i 

i  i  i 

_  4dA     n  -f-  «2        4^2     «  +  3«2-f-2«5 
n  2  n*  6 

Multiplying   this  by  --  and  by  — ,  we   have   the  approximate 
volume  of  the  spherical  segment, 


latter  part  of  the   nineteenth   century   received   the  Kenkon  no  Maki  (possibly 
another   name   for  the   Kohai  no   Ri)  from   their   teachers,   as   Uchida  Gokan 
told  N.  OKAMOTO  and  as  we  are  assured  by  T.  HAGIWARA. 
1  See  page  155,  note  4. 


158  VIII.  The  Yenri  or  Circle  Principle. 


6 

of  which  the  limit  for  n  =  °°  is 


The  same  general  method  appears  in  the  writings  of  Matsunaga, 
Yamaji,  and  others. 

It  has  already  been  stated  that  Isomura  and  Takebe  found 
the  spherical  surface  by  means  of  the  difference  of  volumes 
of  two  concentric  spheres.  In  this  work  the  same  thing  is 
done  for  the  surface  of  an  ellipsoid.  The  volume  of  the  solid 

is  given  as  —  ,.  —  >  but  with  no  proof.    Another  ellipsoid  is  taken 

with   axes   a  +  2k   and    b  +  2k,    and    the    difference   of  their 
volumes  is  divided  by  k,  giving 

—  (-tab  +  b*  +  2ak  +  $bk  +  4/£2), 

<J 

the  limit  of  which,  for  k  =  o,  is 

y  (2  ad  +  b*}. 

This  treatment  is  an  improvement  upon  that  of  Isomura  and 
Takebe  because  it  is  general  rather  than  numerical.  We  there- 
fore have  here  a  further  development  of  the  yenri,  in  which 
it  takes  on  a  little  more  of  the  nature  of  the  Western  calculus, 
but  still  in  only  a  narrow  fashion. 

In  the  same  way,  little  by  little,  some  progress  was  made 
in  the  use  of  infinite  series.  Takebe's  series  for  the  circular 
arc  appears  again  in  1739  in  a  work  entitled  Hoy  en  Sankyd? 
written  by  Matsunaga  Ryohitsu,2  who  received  the  secrets  of 
the  Seki  school  from  Araki,  under  whom  he  had  studied.  The 
Araki-Matsunaga  school,  while  it  started  under  a  less  brilliant 
leader  than  the  school  of  Takebe,  became  the  more  prosperous 

1  Literally,  Mathematical  Treatise  on  Polygons  and  Circles. 

2  His  former  name  was  Terauchi  Gompei.     He  is  also  known  as  Matsunaga 
Yoshisuke. 


VIII.  The  Yenri  or  Circle  Principle.  159 

as  time  went  on,  and  seems  to  have  inherited  most  of  Seki's 
manuscripts.  Araki,  indeed,  gave  the  name  to  Seki's  Seven 
Books,1  and  upon  his  death  in  17 18,2  at  the  age  of  seventy- 
eight,  he  could  look  back  upon  intimate  associations  with  the 
mathematics  of  the  past,  and  upon  the  renaissance  in  the  labors 
of  Seki,  and  could  anticipate  a  fruitful  future  in  the  promise 
of  Matsunaga. 

Matsunaga  was  born  at  Kurume  in  Kyushu,  or  possibly  in 
Terauchi  in  Awari.  His  given  name  being  Terauchi  Gompei, 
we  find  some  of  his  works  signed  with  the  name  Terauchi. 
He  served  under  Naito  Masaki,  Lord  of  Taira  in  Iwaki  and 
afterward  Lord  of  Nobeoka  in  Kyushu,  himself  no  mean  mathe- 
matician. Indeed  it  was  he  whose  insistence  led  Matsunaga  to 
adopt  the  name  tenzan  for  the  Japanese  algebra,  replacing  the 
name  Kigen  seiho  as  used  by  Seki.  Matsunaga  was  a  prolific 
writer 3  and  it  is  to  him  that  the  perpetuation  of  the  doctrines 
of  the  master,  under  the  title  "School  of  Seki",  was  due.  He 
died  in  the  sixth  month  of  1744.* 

In  the  statutes  of  the  school  of  Seki,  as  laid  down  by  him, 
the  work  was  arranged  in  five  classes,  Seki  himself  having 
arranged  it  in  three.  The  two  upper  classes  were  termed 
Betsnden  and  Inka,5  the  latter  covering  Seki's  Seven  Books, 
and  being  open  only  to  one  son  of  the  head  of  the  school 
and  to  two  of  the  most  promising  pupils.  These  three  initiates 
were  required  to  take  a  blood  oath  of  secrecy,6  and  still  further 

1  The  Seki-ryu  Shichibtisho,  published  at  Tokyo  as  a  memorial  volume  on 
the  two  hundredth  anniversary  of  Seki's  death.  See  also  ENDO,  Book  II, 
p.  42.  There  is  some  doubt  as  to  the  titles  of  the  seven  books. 

a  C.  KAWAKITA  in  the  Honcho  Siigaku  Koenshu,  p.  I. 

3  His  works  include    the   following:   Danti  Shosa  (1716),  Embi  Empi  Ryo- 
jutsu  (1735),    Horo    Yosan,    Hoyen   Sankyo  (1739),    Hoyen   Zassan,   Kaiko   Un-o 
(1747,  posthumous),  AT/'o  Tokusho,  Sampo  S'Ausei,  Sampd  Tetsujulsu. 

4  As  stated  in  a  manuscript  by  Hagiwara. 

5  These  names  may  possibly    mean   "Special   Instruction"   and   "Revealed 
by  Swearing."    One  who  completed  these  classes  received  the  two  diplomas 
known  as  Belsuden-menkyo  and  Inka-menkyo. 

6  ENDO,  Book  II,  p.  82  seq.     On  the  five  diplomas  see  also  HAYASHI,  T., 
The  Fukitdai  and  Determinants  in  Japanese  Mathematics,  in  the  Tokyo  STigakii- 


l6o  VIII.  The  Yenri  or  Circle  Principle. 

analogy  to  the  ancient  Pythagorean  brotherhood  is  seen  in  the 
mysticism  of  the  founder.  Matsunaga  writes1  as  Pythagoras 
might  have  done:  "Reason  is  determinate,  but  Spirit  wanders 
in  the  realm  of  change.  Where  Reason  dwelleth,  there  is 
Number  found;  and  wheresoever  Spirit  wanders,  there  Number 
journeys  also.  Spirit  liveth,  but  Reason  and  Number  are 
inanimate,  and  act  not  of  their  own  accord.  The  way  whereby 
we  attain  to  Number  is  called  The  Art.  Heaven  is  independent, 
but  wherever  there  are  things  there  is  Number.  Things, 
Number, — these  are  found  in  nature.  What  oppresses  the 
high  and  exalts  the  humble;  what  takes  from  the  strong  and 
gives  to  the  weak;  what  causes  plenty  here  and  a  void  there; 
what  shortens  that  which  is  long  and  lengthens  that  which  is 
short;  what  averages  up  the  excess  with  the  defect,  — this  is 
the  eternal  law  of  Nature.  All  arts  come  from  Nature,  and 
by  the  Will  alone  they  cannot  exist." 

Matsunaga's  Hoyen  Sankyo  is  composed  of  five  books,  and 
is  devoted  entirely  to  formulas  for  the  circumference  and 
arcs  of  a  circle,  no  analyses  appearing.2  His  first  series  is  as 
follows: 

J^  =  T  +-1L.+      l*'22     4-      T2'22-32       ,   . 
9  3-4         3.4.5.6         3-4-S-6. 7-8 

This  is  followed  by 

?L  ==  i  +  _!!_  +      I2'32       ,   1      ^3M1_  .   ,   . 
3  4.6         4.6.8.10         4.6.8.10.12.14 

a  series  which  is   then    employed    for   the  evaluation  of  TT  to 
fifty  figures.     The  result  is  the  following: 
71=3.14159  26535  89793  23846  26433  83279  50288  41971 
69399  5751- 

Buturigakkwai  Kizi,  vol.  V  (2),  no.  5,  1910.  Yamaji  seems  to  have  revealed 
the  secrets  to  three  besides  his  son. 

1  Hoyen  Sankyo,  1739.    This  work  may  have  been  closely  connected  with 
the  anonymous  Kohai  Shokai. 

2  We  are  informed  by  N.  OKAMOTO  that  Uchida  Gokan  used  to  say  that 
the  original  manuscripts  containing  the  analyses  were  burned  purposely  after 
the   work   was   finished.     Matsunaga's   Hoyen   Zassan    (Miscellany   concerning 
Regular  Polygons  and  the  Circle)  is  now  unknown. 


VIII.  The  Yenri  or  Circle  Principle.  l6l 

The  same  value  is  given  in  the  Hoyen  Kiko,  written  by  Lord 
Arima  in  1766,  together  with  the  numerical  calculations  involved. 
The  value  was  first  actually  printed  in  the  SJiTiki  Sampo,  written 
by  Arima  under  an  assumed  name,  in  1769. 

Matsunaga  next  gives  Takebe's  series  for  the  square  of  an 
arc,1  this  being  followed  by  three  series  for  the  length  of  an 
arc  a  with  chord  c  as  follows: 


.4/AY        2.4.6/A\3 

W  +3:5:7(7) 


-^fi     ±.(*\_.  2  (AY_  2-4  f*v_  ..1 
~L       s   V77    3.5^"  s^yw        J 

The  series  for  the  altitude  //  in  terms  of  the  arc  is 


2    Zd  V  '       (2«)! 

and  for  the  chord  c  it  is 

03  a5  a7 

~  2.3^+  2.3.475^4  ~  2.3.4.5.6.7^    ' 

which  is  at  once  seen  to  be  a  form  of  the  series  for  sin  a.2 
The  area  s  of  a  circular  segment  is  given  as 

s  13 


where  c  =  chord   of  the  arc,  d  =  diameter  of  the  circle,  and 
h  =  height  of  the  segment. 

Matsunaga  also  gives  some  interesting  formulas  for  com- 
puting the  radius  x  of  a  circle  circumscribed  about  a  regular 
polygon  of  n  sides,  one  side  being  s,  and  for  computing  the 
apothem. 

1  Which   appeared   in   the  Yenri  Kohai-jutsu  and   the  Fnkyu  Te/su-ju/su   of 
Takebe  and  the  Yenri  Hakki  of  Oyama. 

2  These  two  series  appear  in  the  Shuki  Sampo. 

3  The  above  series  are  given  in  the  Hoyen  Sankyb,  Book  I. 

1  1 


1 62  VIII.  The  Yenri  or  Circle  Principle. 

He  also  gives J  formulas  for  the  side  of  the  inscribed  polygon 
in  terms  of  the  diameter  of  the  circle,  for  the  various  diagonals, 
for  the  lines  joining  the  mid-points  of  the  diagonals  and  the 
various  vertices  or  the  mid-points  of  the  sides,2  and  so  on, 
none  of  which  it  is  worth  while  to  consider  in  a  work  of  this 
nature. 

It  will  be  seen  that  the  yenri  as  laid  down  by  Takebe  was 
extended  to  include  solid  figures  treated  somewhat  after  the 
manner  of  Cavalieri,  but  that  it  was  little  more  than  a  rather 
primitive  method  of  using  infinite  series  in  the  measurement 
of  the  simplest  curvilinear  figures  and  the  sphere.  We 
shall  see,  however,  that  it  gradually  unfolds  into  something 
more  elaborate,  but  that  it  never  becomes  a  great  method, 
remaining  always  a  set  of  ingenious  devices. 

1  Hoyen  Sankyo,  Book  III. 

2  Lines  known  as  the  Kyomen-shi. 


CHAPTER  IX. 
The  eighteenth  Century. 

We  have  already  spoken  of  the  closing  labors  of  Seki  Kowa, 
who  died  in  1708,  and  of  Takebe  Kenko  and  Araki,  and  in 
Chapter  X  we  shall  speak  of  Ajima  Chokuyen.  There  were 
many  others,  however,  who  contributed  to  the  progress  of 
mathematics  from  the  time  when  Takebe  made  the  yenri  known 
to  the  days  when  Ajima  gave  a  new  impulse  to  the  science, 
and  of  these  we  shall  speak  in  this  chapter.  Concerning  some 
of  them  we  know  but  little,  and  concerning  certain  others  a 
brief  mention  of  their  works  will  suffice.  Others  there  are, 
however,  who  may  be  said  to  have  done  a  work  that  was  to 
that  of  Seki  what  the  work  of  D'Alembert  and  Euler  was  to  that 
of  Newton.  That  is  to  say,  the  periods  in  Japan  and  Europe 
were  somewhat  analogous  in  a  relative  way,  although  the 
breadth  of  the  work  in  the  two  parts  of  the  world  was  not 
on  a  par.  In  some  respects  the  period  immediately  following 
Seki  was,  save  as  to  Takebe's  work,  one  of  relative  quiet,  of 
the  gathering  up  of.  the  results  that  had  been  accomplished 
and  of  putting  them  into  usable  form,  or  of  solving  problems 
by  the  new  methods.  In  the  history  of  mathematics  such  a 
period  usually  and  naturally  follows  an  era  of  discovery. 

So  we  have  Nishiwaki  Richyu  publishing  his  Sampo  Tengen 
Roku  in  1714,  setting  forth  in  simple  fashion  the  "celestial 
element"  and  the  ycndan  algebra.1  In  1722  Man-o  Tokiharu 
published  his  Kiku  Bunto  S/in,  in  which  he  treated,  among 
other  topics,  the  spiral.  In  1715  Hozumi  Yoshin  published  his 

1  ENDU,  Book  II,  pp.  57,  59. 

IT* 


1 64 


IX.  The  eighteenth  Century. 


Kagaku  Sampo,  the  usual  type  of  problem  book.  In  1716 
Miyake  Kenryu  published  a  similar  work,  the  Guivo  Sampo. 
He  also  wrote  the  Sliojutsu  Sangaku  Znye,  of  which  an  edition 
appeared  in  1795  (Fig.  32).  In  this  he  seems  to  have  had 
some  idea  of  the  prismatoid  (Fig.  33).  In  1718  Ogino  Nobu- 
tomo  wrote  a  work,  the  Kiku  Gempo  Chokcn,  that  has  come 
down  to  us  in  nine  books  in  manuscript  form, — a  very  worthy 


Fig.  32.     From  Miyake  Kenryu's  Shojutsu  Sangaku  Zuye  (1795  edition). 

general  treatise.  Inspired  by  Hozumi  Yoshin's  work,  Aoyama 
Riyei  published  his  Cliugaku  Sampo  in  1719,  solving  the 
problems  of  the  Kagaku  Sampo  and  proposing  others.  These 
latter  were  solved  in  turn  by  Nakane  Genjun  in  his  Kanto 
Sampo  (1738),  by  Nakao  Seisei  in  his  Sangaku  Bemmo,  and 
by  Iriye  Shukei  in  his  Tangen  Sampo  (1759).  Mention  should 
also  be  made  of  an  excellent  work  by  Murai  Mashahiro,  the 
RyocJii  SJiinan,  of  which  the  first  part  appeared  in  1732.  The 
work  was  a  popular  one  and  did  much  to  arouse  an  interest 


IX.  The  eighteenth  Century. 


i65 


L  k&V«S#H 
A  g*f|5f$£i 

r^'ftS    ^|?^ 


-  33-     From  Miyake  Kenryu's  Shojulsu  Sangaku  Zuye  (1795  edition). 


1 66  IX.  The  eighteenth  Century. 

in  the  new  mathematics.  The  problems  proposed  by  Nakane 
Genjun  were  answered  by  Kamiya  Hotei  in  his  Kaisho  Sampo 
(1743),  by  Yamamoto  Kakuan  in  his  Sanzui,  and  by  others. 
To  the  same  style  of  mathematics  were  devoted  Yamamoto's 
Yokyoku  Sampo  (1745)  and  Keiroku  Sampo  (1746),  Takeda 
Saisei's  Sembi  Sampo  (1746),  Imai  Kentei's  Meigen  Sampo 
(1764),  and  various  other  similar  works,  but  by  the  close  of 
the  eighteenth  century  in  Japan,  as  elsewhere,  this  style  of 
book  lost  caste  as  representing  a  lower  form  of  science  than 
that  in  which  the  best  type  of  mind  found  pleasure.  Mention 
should  also  be  made  of  Baba  Nobutake's  Shogaku  Tcnmon  of 
1706,  a  well-known  work  on  astronomy,  that  exerted  no  little 
influence  at  this  period  (Fig.  34). 

Of  the  writers  of  this  general  class  one  of  the  best  was 
Nakane  Genjun  (1701  — 1761),  whose  Kanto  Sampo  (1738) 
attracted  considerable  attention.  His  father,  Nakane  Genkei 
(1 66 1  — 1733),  was  born  in  the  province  of  Omi,  and  studied 
under  Takebe.  He  was  at  one  time  an  office  holder,  but  in 
earlier  years  he  practiced  as  a  physician  at  Kyoto.  His  taste 
led  him  to  study  mathematics  and  astronomy  as  well,  and  he 
seems  to  have  been  a  worthy  instructor  for  his  son,  who  thus 
received  at  second  hand  the  teachings  of  Seki's  greatest  pupil. 
Some  interesting  testimony  to  his  standing  as  a  scholar  is 
given  in  a  story  related  of  a  certain  feudal  lord  of  the 
Kyoho  period  (1716 — 1736),  who  asked  a  savant,  one  Shinozaki, 
who  were  his  most  celebrated  contemporaries.  Thereupon 
the  savant  replied:  "Of  philosophers,  the  most  celebrated  are 
Ito  Jinsai  and  Ogyu  Sorai;  of  astronomers,  Nakane  Genkei 
and  Kurushima  Kinai;1  in  calligraphy,  Hosoi  Kotaku  and 
Tsuboi  Yoshitomo ;  in  Shintoism,  Nashimoto  of  Komo ;  in  poetry 
Matsuki  Jiroyemon;  and  .as  an  actor,  Ichikawa  Danjyuro.  Of 
these,  Nakane  is  not  only  versed  in  astronomy,  but  he  is 
eminent  in  all  branches  of  learning."2 

Nakane  the  Elder  also  published  several  astronomical  works, 

1  Or  Kurushima  Yoshita. 

*  K.  KANO'S  article  in  the  Honcho  Sitgaku  Koenshu,  1908,  p.  II. 


IX.  The  eighteenth  Century. 


I67 


Fig.  34.     From  Baba  Nobutake's  Shogaku  Tenmon  (1706). 


and    composed    a    treatise    in    which   a    new    law   of    musical 
melodies  was  set  forth.1     Through  the  Chinese  works  and  the 


1  This  was   the   Ritsugen   Hakki,   a  work   on  the   description  of  measures. 


l68  IX.  The  eighteenth  Century. 

writings  and  translations  of  the  Jesuit  missionaries  in  China 
he  was  familiar  with  the  European  astronomy,  and  he  re- 
cognized fully  its  superiority  over  the  native  Chinese  theory.  He 
was  prominent  among  those  who  counseled  the  Shogun  Yoshi- 
mune  to  remove  the  prohibition  against  the  importation  and 
study  of  foreign  books,  and  by  order  of  the  latter  he  is  said  to 
have  translated  Mei  Wen-ting's  Li-suan  Ch'iian-shu.'1  In  1711  he 
was  given  a  post  in  the  mint  at  Osaka,  and  in  1721  became  con- 
nected with  the  preparation  of  the  official  calendar.2  In  pure 
mathematics  he  wrote  but  one  work  that  was  published,  the 
Shichijo  Beki  Yenshiki?  although  by  all  testimony  he  was  an 
able  mathematician.  One  of  his  solutions,  appearing  in  Takebe's 
Fukyn  Tetsu-jutsu  (1722),  is  that  of  an  interesting  indeterminate 
equation.  The  problem  is  to  find  the  sides  of  a  triangle  that 
shall  have  the  values  ;/,  n  +  i,  and  n  +  2,  and  such  that  the 
perpendicular  upon  the  longest  side  from  the  opposite  vertex 
shall  be  rational.  Nakane  solves  it  as  follows: 

When  the  sides  are  I,  2,  3,  the  perpendicular  is  evidently 
zero. 

Taking  the  cases  arising  from  increasing  these  values  suc- 
cessively by  unity,  the  following  triangles  satisfy  the  conditions: 

3  13    Si    193 

4  14    52    194 

5  15    53    195 

If  we  represent  these  values  by  a^b^  c^;  a2,  b2,  c2;  a3,  b^  c3;  .  .  ., 
it  will  readily  be  seen  that 


and  similarly  for  the  <$'s  and  c's,  and  hence  we  have  the 
required  solution.  Whether  or  not  he  made  the  induction 
complete  does  not,  however,  appear. 


1  See  page  19.     The  work  is  in  the  library  of  the  Emperor. 

2  For  this   purpose  he   spent  half  of  his   time   in  Yedo,   the   rest   beim 
spent  in  Kyoto. 

3  It  was  printed  in   1691  and  reprinted  in  1798. 


IX.  The  eighteenth  Century.  169 

It  is  also  related  that  Takebe  was  asked  in  1729,  by  the 
Shogun  Yoshimune,  for  the  solution  of  a  certain  problem  on 
the  calendar.  Takebe,  recognizing  the  great  ability  of  the 
aged  Nakane,  asked  him  to  undertake  it;  but  he,  feeling  the 
infirmities  of  his  years,  passed  it  in  turn  to  his  son,  Nakane 
Genjun.  The  result  was  a  new  method  of  solving  numerical 
higher  equations  by  successive  approximations  that  alternately 
exceed  and  fall  short  of  the  real  value,  a  method  that  was 
embodied  in  the  Kaiho  Yeijiku-jutsu*  written  by  Nakane 
Genjun  in  1729.  The  problem  proposed  by  the  Shogun  is  as 
follows:2  "There  are  two  places,  one  in  the  south  and  one  in 
the  north,  from  which  the  elevation  of  the  pole  star  above 
the  horizon  is  36°  and  4O°75'  respectively.  At  noon  on  the 
second  day  of  the  ninth  month  in  a  certain  year  the  shadows 
of  rods  0.8  of  a  yard  high  were  0.59  of  a  yard  and  0.695  of 
a  yard,  respectively,  and  at  the  southern  station  the  center  of 
the  sun  was  36° 37'  distant  from  the  zenith  at  noon  on  the 
da\~  of  the  equinox.  Required  from  these  data  to  determine 
the  ratio  of  the  diameter  of  the  sun's  orbit  to  the  diameter  of 
the  earth,  considering  the  two  to  be  concentric." 

The  solution  of  this  problem  is  too  long  to  be  given  here, 
but  that  of  another  one  in  the  same  manuscript  may  serve  to 
illustrate  Nakane's  methods.  "Given  a  circle  in  which  are 
inscribed  two  equal  smaller  circles  and  another  circle  which 
we  shall  designate  as  the  middle  circle.  Each  of  these  four 
circles  is  tangent  to  the  other  three;  the  difference  of  area 
between  the  large  circle  and  the  three  inscribed  circles  is  120, 
and  the  diameters  of  the  middle  and  small  circles  differ  by  5. 
Required  to  find  the  diameters." 

Nakane  lets  /,  m,  s,  stand  for  the  respective  diameters  of 
the  large  circle,  middle  circle,  and  small  circles. 

Then  s  +  5  =  ni 

and  (s  +  ni)z  —  s2  =  a2,  an  arbitrary  abbreviation. 


1  Literally,  Method  of  Increase  and  Decrease  in  the  Evolution  of  Equations. 

2  From  a  manuscript  of  1729. 


I/O  IX.  The  eighteenth  Century. 


T-I  7        (<*  + 

Then  /  =  v 


and  /2  —  2s2  —  m2  =  102  :  —  • 

4 

He  then  assumes  that     .^  =  7.5, 

whence,  from  the  above,  the  two  sides  of  the  equation  become 

150.0654  and  152.788, 
their  difference,  d^,  being  2.723. 
He  next  tries  s2  =  7.6, 

whence,  as  before,  d2  =  —  0.37811. 

He  then  takes  s3  =  st  +  d    *       =*  7.5878, 


whence  as  before,       —  d^  =  —0.028246. 
He  now  proceeds  as  before,  taking 

,4  =  ,2-  --  =  7-5868..., 


S2  —  S3 

and  in  the  same  way  he  continues  his  approximations  as  far 
as  desired. 

Not  only  did  Nakane  the  younger  study  with  his  father,  but 
he  also  went  to  Yedo  (Tokyo)  to  learn  of  Takebe  and  of 
Kurushima.  Returning  to  Osaka  he  succeeded  his  father  in  the 
mint,  and  in  1738  he  published  the  Kanto  Sampo  followed  in 
1741  by  an  arithmetic  for  beginners  under  the  title  Kanja  Otogi 
Zos/ii.*  In  this  latter  work  the  mercantile  use  of  the  Soroban 
is  explained  (Fig.  35)  and  the  check  by  the  casting  out  of 
nines  is  first  used  in  multiplication,  division,  and  evolution  in 
Japan.  He  died  in  1761  at  the  age  of  sixty. 

The  most  distinguished  of  Nakane  Genkei's  pupils  was  Koda 
Shin-yei,  who  excelled  in  astronomy  rather  than  in  pure 

1  Literally,  A  Companion  Book  for  Arithmeticians. 


IX.  The  eighteenth  Century. 


I/I 


Fig-  35-     From  Nakane  Genjun's  Kanja  Otogi  Zoshi  (1741). 

mathematics,  and  who  died  in  1758.  Among  Koda's  pupils 
were  Iriye  Shukei,  Chiba  Saiyin  (c.  1770),  and  Imai  Kentei 
(1718 — 1780).  Imai  Kentei,  who  left  several  unpublished  manu- 


172  IX.  The  eighteenth  Century. 

scripts,  had  as  his  most  prominent  pupil  Honda  Rimei  (1751 — 
1828),'  a  man  of  wide  learning  and  of  great  influence  in  edu- 
cation. Honda  numbered  among  his  pupils  many  distinguished 
men,  including  Aida  Ammei,  Murata  Koryu,  Kusaka  Sei,  Mogami 
Tokunai,  Sakabe  Kohan,  and  Baba  Seitoku.  He  gave  much 
attention  to  the  science  of  navigation  and  to  public  affairs,  and 
even  advocated  the  opening  of  Japan  to  foreign  trade.  He 
was  familiar  with  the  Dutch  language,  and  made  some  attempt 
at  mathematical  research,2  and  to  his  influence  Mamiya  Rinzo, 
the  celebrated  traveler,  acknowledged  his  deep  indebtedness. 

Another  prominent  disciple  of  Takebe's  was  Koike  Yiiken 
(1683 — 1754),  a  samurai  of  Mito,  where  he  presided  over  the 
Shokokivan  or  Institute  for  Historical  Research.  By  order  of 
his  lord  he  went  to  Yedo  and  learned  mathematics  from  Takebe, 
acquiring  at  the  same  time  some  knowledge  of  astronomy. 

His  successor  in  the  SJiokokivan  at  Mito  was  Oba  Keimei 
(1719—1785),  but  neither  one  contributed  anything  to  mathe- 
matics beyond  a  sympathetic  interest  in  the  progress  of  the 
science. 

Among  the  pupils  of  Nakane  Genjun,  and  therefore  of  the 
Takebe  branch  of  the  Seki  school,  was  Murai  Chuzen,  a  Kyoto 
physician.  He  wrote  a  work  entitled  the  KaisJio  Tempei  Sampo  $ 
(1765)  which  treated  of  the  solution  of  numerical  higher 
equations.  Three  years  later  one  of  his  pupils,  Nagano  Seiyo, 
published  a  second  part  of  this  work  in  which  he  attempted 
to  explain  the  methods  employed  in  the  solutions.  For  example, 
Murai4  takes  the  equation 

6726  —  373  #  +  #2  =  o. 
He  then  finds  the  relation 

373  -372.1  =  i, 

1  Also  known  as  Honda  Toshiaki. 

2  OZAWA,  Lineage  of  mathematicians  (in  Japanese),  and  the  epitaph  on  Honda's 
tomb. 

3  Literally,  The  Posting  of  Soldiers  in  the  Evolution  of  Equations. 

4  ENDO,  Book  II,  pp.  137 — 139. 


IX.  The  eighteenth  Century. 


173 


and  multiplies  the  372  into  the  absolute  term  (6726)  and  then 
subtracts   373    as    often  as  possible,  leaving  a  remainder  36 1.1 
This  remainder  is  added  to  6726  and  the  result  is  divided  by 
373,  the  quotient,  19,  being  a  root. 
Similarly,  in  the  equation 

—  25233  —  2284^  +  25^3  =  o, 
Murai  claims  first  to  take  the  relation 

2284  x  1 1  —  25  m  =  —  i, 

and  states  that  he  multiplies  1 1    into  the   absolute  term,  sub- 
tracting 2284  from  the  product  until  he  reaches  a  remainder, 
which   is  the  root  required,  a  process  that  is  not  at  all  clear. 
Of  course  the  method  is  not  valid,  for  in  the  equation 

xz  —  %x  +  15  =o 

it  gives   2  instead    of  3  or  5   for  the  root.     Murai  must  have 
been  aware  that  his  rule  was  good  only  for  special  cases,  but 


Fig-  36.     From  Murai  Chflzen's  Sampo  Ddshl-mon  (1781). 


1  Briefly,    372X6726  =  2,502,072,   and   2,502,072-^-373  =  6707    with    a 
remainder  361. 


174 


IX.  The  eighteenth  Century. 


Fig.  37-     The  Pascal  triangle  as  given  in  Murai's 
Sampo  Doshi-mon  (1781). 

he  makes  no  mention  of  this  fact.  Nevertheless  he  assisted 
in  preparing  the  way  for  modern  mathematics  by  discouraging 
the  use  of  the  sangi,  which  were  already  beginning  to  be  looked 
upon  as  unwieldy  by  the  best  algebraists  of  his  time. 
..  Murai  also  wrote  a  Sampo  Dos/ii-mon,  or  Arithmetic  for  the 
Young  (see  Figs.  36 — 38),  which  was  intended  as  a  sequel 


IX.  The  eighteenth  Century. 


175 


Fig.  38.     From  Murai's  Sampo  Doshi-mon  (1781).       /M. 

to  the  Kaiija  Otogi  Zoski  of  Nakane  Genjun.  The  work- 
appeared  in  1781,  and  contains  numerous  interesting  pictures 
of  primitive  work  in  mensuration  (Fig.  36),  and  the  Pascal 


1/6  IX.  The  eighteenth  Century. 

triangle  (Fig.  37).  It  is  also  noteworthy  because  of  its  treat- 
ment of  circulating  decimals.  The  problem  as  to  the  number 
of  figures  in  the  recurring  period  of  a  unit  fraction  was  first 
mentioned  in  Japan  by  Nakane  in  his  Kanto  Sampo  (1738) 
and  solutions  of  an  unsatisfactory  nature  appeared  in  Ikebe's 
KaisJw  Sampo  (1743)  and  in  Yamamoto's  Sansid  (1745).  Na- 
kane's  writings  upon  the  problem  were  no  longer  extant,  so 
that  Murai  had  practically  the  field  before  him  untouched, 
although  he  really  did  little  with  it.  His  theory  is  brief,  for 
he  first  divides  9  by  2,  3,  ...  9,  getting  the  figures  45,  3,  225, 
18,  15,  x  (not  divisible),  1125,  I, — without  reference  to  the 
decimal  points.  He  then  concludes  that  if  unity  is  divided  by 
45>  3>  225>  •  •  •>  the  result  will  have  one-figure  repetends.  Simil- 
arly he  divides  99  by  2,  3,  ...  9,  getting  the  figures  495,  33, 
2475,  189,  .  .  .,  and  then  divides  unity  by  these  results,  getting 
two-figure  repetends. 

In  his  explanation  of  the  use  of  the  sorobaii  Murai  gives 
certain  devices  that  his  predecessors  had  not  in  general  used. 
For  example,  in  extracting  the  square  root  he  divides  half  of 
the  remainder  by  the  part  of  the  root  already  found,  which 
he  evidently  thought  to  be  a  little  easier  on  the  soroban  than 
to  divide  by  twice  this  root.  In  treating  of  cube  root  he 
proceeds  in  an  analogous  fashion,  dividing  a  third  of  the 
remainder  twice  by  the  part  of  the  root  already  found.  \Ye 
have  said  that  these  devices  had  not  been  used  in  general 
before  Murai,  but  they  had  already  been  given  by  at  least  one 
writer,  Yamamoto  Hifumi,  in  his  Hayazan  Tebikigusa^  in  1775. 

Contemporary  with  Nakane  Genkei,  and  a  friend  of  his, 
was  a  curious  character  named  Kurushima  Yoshita,  a  native 
of  Bitchu,  at  one  time  a  retainer  of  Lord  Naito,  and  a  man 
of  notorious  eccentricity  and  looseness  of  character.  It  is 
related  of  him  that  when  he  had  to  leave  Kyushu  to  take 
up  his  residence  in  Yedo,  he  used  all  of  his  mathematical 
manuscripts  to  repair  his  basket  trunks  for  the  journey.  He 
must,  however,  have  been  a  man  of  mathematical  ability, 

1  Literally,  Handbook  of  Rapid  Calculations. 


IX.  The  eighteenth  Century.  177 

for  he  was  the  friend  not  only  of  Nakane  but  also  of  Matsunaga, 
and  he  had  at  least  one  pupil  of  considerable  attainments, 
Yamaji  Shuju.  He  died  in  1757.  Among  the  fragments  of 
knowledge  that  have  been  transmitted  concerning  him  is  a 
formula  for  the  radius  r,  of  a  regular  n-gon  of  side  s,  ex- 
pressed in  an  infinite  series.1 

Kurushima  also  knew  something  of  continued  fractions,  since 
in  Ajima's  Fukyu  Sampo2  and  other  works  it  is  shown  how 
he  expressed  a  square  root  in  this  manner,  with  the  method 
of  finding  the  successive  convergents.  This  seems  to  have 
been  an  invention  made  by  him  in  I726.3  It  is  repeated  in  a 
work  written  in  1748  by  Hasu  Shigeru,  a  pupil  of  one  Horiye 
who  had  learned  from  Takebe.  In  the  preface  Horiye  says 
that  the  method  is  one  of  the  most  noteworthy  of  his  time.4 

Kurushima  was  also  interested  in  magic  squares,  and  his 
method  of  constructing  one  with  an  odd  number  ot  cells  is 
worth  repeating. s 

The  plan  may  briefly  be  described  as  follows: 

Let  n   be   the    number   of   cells  in  one  side.     Arrange  the 

1  ENDO,    Book  II,  p.  112;  Kawakita  in   the  Honcho  Sugaku  Koenshu,  p.  6. 
On  the  life  of  Kurushima  there  is  a   manuscript  (Japanese)  entitled  Tea-table 
Stories   told  by   Yamaji.     This   formula   was   first  published  in  Aida   Ammei's 
Sampo   Kokon   Tsitran    (General    View    of    Mathematical    Works    ancient   and 
modern),  1795,   Book  VI.     It  appears    again  in   Chiba's   Sampo  Shinsho  (New 
Treatise  on  Mathematical  Methods).    See  FUKUDA,  Sampo  Tamatebako.    Book  II, 
p.  33;    ENDO,  Book  III,   p.  33.     Kurushima    also  wrote  the  Kyushi  Kohai  So 
(Incomplete  Fragments  on  the  arc   of    a   circle)   in   which   he   treated   of  the 
minimum  ratio  of  an  arc   to  its  altitude.    It  exists  only  in  manuscript.     In  it 
is  also  some  work  in  magic  cubes. 

2  In  manuscript,  compiled  by  Kusaka. 

3  Possibly  Takebe    was    the    first  Japanese  to  employ  continued  fractions, 
in  his  Fukyu  Telsujutsu  (1722).     See   also   the  Taisei  Sankyo,  where  they  are 
found.    But  their  application  to  square  root  begins,  in  Japan,  with  Kurushima. 
C.  KAWAIUTA   relates  in   the    Siigaku  Ilotki   that   this  was  done   in   the    first 
month  of  1726. 

4  HORIYE'S   preface   to  HASU'S  Heiho  Reiyaku  Genkai,   1748,   in  manuscript. 
See  also  ENDO,  Book  II,  p.  105. 

5  It  is  given  in  his  manuscript  Kyushi  Iko  (Posthumous  Writings  of  Kuru- 
shima), Book  I. 

12 


I78 


IX.  The  eighteenth  Century. 


numbers   I,  n2,  n,  and  k  =  n2  +  i  — n  as   in  the  figure.     Then 
take  --  (n2  +  i)  as   the   central  number,   and   from  this,  along 


n 


n1 


k 


D 


B 


CD,  arrange  a  series  decreasing  towards  C  and  increasing 
towards  D  by  the  constant  difference  n.  Next  fill  the  cells 
along  the  oblique  lines  through  n  and  n2,  and  through  i  and 
k,  according  to  the  same  law.  Now  fill  the  cells  along  AB 
and  the  two  parallels  through  n  and  i,  and  through  n*  and  k, 
by  a  series  decreasing  towards  A  and  increasing  towards  B  by 
the  constant  difference  i.  The  rest  of  the  rule  will  be  apparent 
by  examining  the  following  square: 


22 

47 

16 

4' 

10 

35 

4 

5 

23 

48 

17 

42 

ii 

29 

30 

6 

24 

49 

18 

36 

12 

13 

3f 

7 

25 

43 

19 

37 

38 

H 

32 

i 

26 

44 

20 

21 

39 

8 

33 

2 

27 

45 

46 

15 

40 

9 

34 

3 

28 

IX.  The  eighteenth  Century.  179 

It  is  also  worthy  of  note  that  Kurushima  discussed1  the 
problem  of  finding  the  maximum  value  of  the  quotient  of  the 
altitude  of  a  circular  segment  by  its  arc.  In  this  there  arises 
the  equation 

.r  4  .r  6  r  S 


3.6         3.5X6.8          3.5.7X6.8.10 

3.5.7-9X6.8.  10.12  "* 

He  speaks  of  this  as  an  "unlimited  equation",  and  after  a 
complicated  solution  he  reaches  the  result, 

*•=  5. 4341 3 1 504304. 

Mention  should  also  be  made  of  a  value  of  u2  given  by 
Kurushima,  —  —„„  ;  but  his  method  of  obtaining  it  is  not  known.2 

In  the  first  half  of  the  eighteenth  century  there  lived  in 
Osaka  one  Takuma  Genzayemon,  concerning  whose  life  and 
early  training  we  know  practically  nothing.  Some  have  said 
that  he  learned  mathematics  in  the  school  of  Miyagi,  but  all 
that  is  definitely  known  is  that  he  established  a  school  in 
Osaka.  He  is  of  interest  because  of  his  work  upon  the  value 
of  IT,  a  problem  that  he  attacks  in  the  Dutch  manner  of  a 
century  earlier.  He  seems  to  have  been  the  only  mathe- 
matician in  Japan  who  used  for  this  purpose  the  circumscribed 
regular  polygon  as  well  as  the  inscribed  one  of  a  large  number 
of  sides.  He  bases  his  conclusions  upon  the  perimeters  of 
polygons  of  17,592,186,044,416  sides  which  he  stated  to  be 

3.14159     26535     89793     23846     26433     6658, 
3.14159     26535     89793     23846     26434     67. 

He  takes  the  average  of  these  numbers,  and  thus  finds  the 
value  correct  to  twenty-five  figures.  It  is  related  that  this 
was  looked  upon  as  one  of  the  most  precious  secrets  of  his 


1  In  his  manuscript  entitled  Kyiishi  Kohai-so. 

2  ENDO,  Book  II,   p.  127.     It   is    found   in   manuscript   in   the   posthumous 
writings  of  Kurushima. 


l8o  IX.  The  eighteenth  Century. 

school.1  The  most  distinguished  of  Takuma's  followers  was 
Matsuoka  Noichi  (or  Yoshikadsu),  who  published  a  very  usable 
textbook  in  1808,  the  Sampo  Keiko  Taizen? 

Mention  has  already  been  made  of  Matsunaga  Ryohitsu,* 
but  his  work  is  such  as  to  merit  further  notice.  One  of  his 
most  important  treatises  is  embodied  in  a  manuscript  called 
the  Sampo  Shusei*  consisting  of  nine  books  of  which  the  first 
five  are  devoted  to  indeterminate  analysis  as  applied  to  questions 
of  geometry.  He  considers,  for  example,  the  Pythagorean 
triangle  of  sides  a,  b,  and  hypotenuse  c,  and  lets 

a  =  2m  +  i,  c  —  b  =  211, 

whence 

,  _  a  —  fr  _      a*      _  (2»  +  i)» 
c  "t~  c/  —  ,    —  .  —  . 

c  —  b         c  —  o  2n 

whence  b  and  c  assume  the  form 


Hence  the  three  sides  may  be  represented  by 

4»  (2m  +  i),  (2m  +  i)2  —  4«2,  (2m  +  i)2  4-  4«2. 

He  also  attacks  the  problem  by  letting  the  perpendicular  p 
from  the  vertex  of  the  right  angle  cut  the  hypotenuse  into 
the  segments  c'  and  c"  .  He  then  gets 

b2  —  a*  =  (c"2  +p2)  —  (c'2  +/2) 

=  (c"  +  c'}  (c"  —  c'}  ==c  (c"  —  c'\ 
2ab  =  c  .  2p, 

and  a2  +  b1  =  c2. 

Then  since         p2  =  c'c",  we  have 

(c"  —  c')2  +  (2/)2  =  c\ 

1  ENDO,  T.,  On  the  development  of  the  mensuration  of  the  circle  in  Japan 
(in  Japanese),  Rigakkai,  Book  III,  no.  4. 

a  Literally,  A  Complete  Treatise  of  mathematical  instruction. 

3  See  page  158.     The  name  also  appears  as  Matsunaga  Yoshisuke. 

4  Literally,  .A  Collected  Treatise  on  mathematical  methods.    It  is  undated. 
His  Hoyen  Sankyo  is  dated  1739  in  one  of  the  prefaces  and  1738  in  another. 


IX.  The  eighteenth  Century.  l8l 

whence  the  sides  of  a  right  triangle  may  be  represented  by 
b2  —  a2,  2  ad,  and  a2  +  b2. 

Matsunaga  was,  like  most  of  his  contemporary  geometers, 
interested  in  the  radius  of  the  regular  polygon  of  n  sides,  each 
side  being  equal  to  s.  His  formula, 

2  =  62370  «4  4- 107480*2  +  83577  _  ^ 
2462268  «•*  —  3857400 

is  claimed  to  give  the  radius  correct  to  six  figures.1  A  more 
complicated  formula,  requiring  the  extraction  of  a  seventh  root, 
is  given  in  Irino  Yosho's  Kakuso  Sampo  (1743),  but  it  is  no 
more  accurate. 

Still  another  formula  of  this  nature  is  given  by  Matsunaga's 
pupil  Yamaji  Shuju  (1704 — 1772)2, 

7-2  =  (15 1 7621639810;^  +  1 0049747  20807  n6 
+  16637450385672*)  s2  ~  (59913200861841  n6 
-  i 5 743 2047 5 80066 «4 +  I355297564732o6«2 
-35692069491815). 

Such  efforts,  however,  are  interesting  chiefly  for  the  same 
reason  as  the  Japanese  ivory  carving  of  spheres  within  spheres, 
— examples  of  infinite  painstaking.  Yamaji  was  a  native  of 
the  province  of  Bitchu,  and  later  he  became  a  samurai  of  the 
shogunate,  serving  as  assistant  in  the  astronomical  department. 
He  first  studied  under  Nakane,  and  upon  Nakane's  leaving 
Yedo  for  Kyoto  he  came  under  the  latter's  friend  Kurushima. 
When  Kurushima  moved  to  Kyushu,  Yamaji  became  a  pupil 
of  Matsunaga.  He  was  thus,  as  he  relates  in  his  Tea-table 
Stories,  privileged  to  know  the  mathematical  secrets  of  three 
of  the  best  teachers  of  Japan.  While  he  was  not  himself  a 
great  contributor  to  the  science,  he  proved  to  be  a  great 
teacher,  so  that  when  he  died  not  a  few  sucessful  mathe- 

1  The  reader  may  consider  it  for  «  =  4,  s  =  \^2,  r=—.     It  is  also  given 
in  Arima's   Hoyen  Kiko  (1766),   but  credit  is  there  given  to  Matsunaga.     See 
also  ENDO,  Book  II,  p.  109. 

2  ARJM.V,  Hoyen  Kiko;   ENDO,  Book  II,  p.  108. 


1 82  IX.  The  eighteenth  Century. 

maticians  were  counted  among  his  pupils,  including  Lord  Arima, 
Fujita,  and  Ajima.  It  is  possible  that  the  Kenkon  no  Maki 
was  written  by  him,  and  also  the  Kohai  no  Ri  and  other 
manuscripts  on  the  yenri,  but  the  Gyokuseki  Skin-jutsu *  is  the 
only  work  of  importance  that  is  certainly  his.  In  this  is  given 
a  treatment  of  the  volume  of  the  sphere  by  a  kind  of  integra- 
tion much  like  that  to  be  found  in  the  anonymous2  Kigenkai. 
Of  Yamaji's  pupils  the  first  above  mentioned  was  Arima  ^ 
Raido  (1714 — 1783),  Lord  of  Kurume  in  Kyushu.  It  was  he, 
it  will  be  recalled,  who  first  published  the  tenzan  algebra  that 
had  been  kept  a  secret  in  the  Seki  school  since  the  days  of 
the  founder.  His  Skuki  Sainpo  in  five  books  was  published  in 
1769  under  the  nom  de  plume  of  Toyota  Bunkei,  possibly  the 
name  of  one  of  his  vassals.  The  work  must  certainly  have 
been  Arima's,  however,  since  only  a  man  in  his  position  would 
have  dared  to  reveal  the  Seki  secret.  In  this  treatise  Arima 
sets  forth  and  solves  one  hundred  fifty  problems,  thus  being 
the  first  noted  writer  to  break  from  the  old  custom  of  solving 
the  problems  of  his  predecessors  and  setting  others  for  those 
who  were  to  follow.  His  questions  related  to  indeterminate 
analysis,  the  various  roots  of  an  equation,  the  algebraic  treat- 
ment of  geometric  propositions,  binomial  series,  maxima  and 
minima,  and  the  mensuration  of  geometric  figures,  including 
problems  relating  to  tangent  spheres  (Fig.  39).  The  curious 
Japanese  manner  of  representing  a  sphere  by  a  circle  with  a 
lune  on  one  side  is  seen  in  Fig.  39.  In  this  work  appears  a 
fractional  value  of  TT, 

=  42822     45933     49304 
13630     81215     70117  ' 

that  is  correct  to  twenty-nine  decimal  places.  Arima  also  wrote 
several  other  works,  including  the  Hoy  en  Kiko  (1766)*  and  the 
Skosa  San-yo  (1764),  but  none  of  these  was  published. 

*  Literally,  The  Exact  Method  for  calculating  the  volume  of  a  sphere. 

2  Or  Yenri  Kenkon  Sho. 

3  Not  Akima,  his  ancestor,  as  is  sometimes  stated. 

4  In  this  is  also  given  the  value  of  TT  mentioned   above,    and   the  powers 
of  ir  from  ir2  to  ir22  for  the  first  thirty-two  figures. 


IX.  The  eighteenth  Century.  183 

Among  the  vassals  of  Lord  Arima  was  a  certain  Honda 
Teiken  (1734—1807),  who  was  born  in  the  province  of  Musashi. 
He  is  known  in  mathematics  by  another  name,  Fujita  Sadasuke, 
which  he  assumed  when  he  came  to  manhood,  a  name 
that  acquired  considerable  renown  in  the  latter  half  of  the 
eighteenth  century.  As  a  youth  he  studied  under  Yamaji,  and 
even  when  he  was  only  nineteen  years  of  age  he  became,  on 


Fig-  39-     From  Arima's  Shuki  Sampo  (1769). 

Yamji's  recommendation,  assistant  to  the  astronomical  depart- 
ment of  the  shogunate.  For  five  years  he  labored  acceptably 
in  this  work,  but  finally  was  compelled  to  resign  on  account 
of  trouble  with  his  eyes.  Arima  now  extended  to  him  a  cordial 
invitation  to  accompany  him  to  Yedo,  whither  he  went  for 
service  every  second  year,  and  to  act  as  teacher  of  arithmetic.1 
Here  he  published  his  Seiyo  Sampo  (1779),  a  work  in  three 
books,  consisting  of  a  well  arranged  and  carefully  selected  set 
of  problems  in  the  tenzan  algebra.  This  book  was  so  clearly 
written  as  to  serve  as  a  guide  for  teachers  for  a  long  time 
after  its  publication.  In  Fig.  40  is  shown  one  of  his  problems 

1  Kawakita,  in  the  Honcho  Siigaku  JCoenshtl,  1908,  p.  8. 


1 84 


IX.  The  eighteenth  Century. 


relating  to  tangent  spheres  in  a  cone.  Fujita  also  published 
several  other  works,  including  the  Kaisei  Tengen  SJiinan  (1792),* 
and  wrote  numerous  manuscripts  that  were  eagerly  sought  by 
the  mathematicians  of  his  time,  although  of  no  great  merit  on 
the  ground  of  originality.  He  died  in  1807  at  the  age  of 
seventy-two  years,  respected  as  one  of  the  leading  mathe- 
maticians of  his  day,  although  he  did  not  merit  any  such 
standing  in  spite  of  his  undoubted  excellence  as  a  teacher. 

Fujita's  son  Fujita  Kagen 
(1765—1821)  was  also  a  mathe- 
matician of  some  prominence. 
He  published  in  1790  his  SJiim- 
pekiSampo  (Mathematical  Prob- 
lems suspended  before  the 
Temple),2  and  in  1806  a  sequel, 
the  Zoku  Shinipeki  Sainpd. 
The  significance  of  the  name 
is  seen  in  the  fact  that  the 
work  contains  a  collection  of 
problems  that  had  been  hung 
before  various  temples  by 
certain  mathematical  devotees 
between  1767  and  the  time 


Fig.  40.     From  Fujita  Sadasuke's 
Seiyo  Sampo  (1779). 


when  Fujita  wrote,  together 
with  rules  for  their  solution.  This 
strange  custom  of  hanging 

problems  before  the  temples  originated  in  the  seventeenth  cen- 
tury, and  continued  for  more  than  two  hundred  years.  It 
may  have  arisen  from  a  desire  for  the  praise  or  approval  of 
the  gods,  or  from  the  fact  that  this  was  a  convenient  means 
of  publishing  a  discovery,  or  from  the  wish  to  challenge  others 
to  solve  a  problem,  as  European  students  in  the  Middle  Ages 
would  post  a  thesis  on  the  door  of  church.  A  few  of  these 


1  We  follow  EndO.     Hayashi  gives  1793. 

2  There  was  a  second  edition  in  1796,  with  some  additions. 


IX.  The  eighteenth  Century.  185 

problems  are   here   translated1    as   specimens  of  the  work   of 
Japanese  mathematicians  at  the  close  of  the  eighteenth  century. 
"There  is    a  circle   in  which   a   triangle   and    three    circles, 
A,  B,   C,  are  inscribed  in  the  manner  shown  in  the  figure. 


Given  the  diameters  of  the  three  inscribed  circles,  required  the 
diameter  of  the  circumscribed  circle."  The  rule  given  may  be 
abbreviated  as  follows: 

Let  the  respective  diameters  be  x,  y,  and  z,  and  let  xy  =  a. 
Then  from  a  2  take  \(x  —y)  z\  .  Divide  a  by  this  remainder 
and  call  the  result  b.  Then  from  (x  +  y)  z  take  a,  and  divide 
0.5  by  this  remainder  and  add  b,  and  then  multiply  by  z 
and  by  a.  The  result  is  the  diameter  of  the  circumscribed 
circle.2  To  this  rule  is  appended,  with  some  note  of  pride,  the 
words:  "Feudal  District  of  Kakegawa  in  Yenshu  Province, 
third  month  of  1795,  Miyajima  Sonobei  Keichi,  pupil  of  Fujita 
Sadasuke  of  the  School  of  Seki." 

Another  problem  is  stated  as  follows:  "Two  circles  are  de- 
scribed, one  inscribing  and  the  other  circumscribing  a  quadri- 


From  the  edition  of  1796. 

That  is 

.  0.5 


r  _  xy^ 
L*Vl  —  [(•*-. 


1 86  IX.  The  eighteenth  Century. 

lateral.  Given  the  diameter  of  the  circumscribed  circle  and 
the  product  of  the  two  diagonals,  required  to  find  the  diameter 
of  the  inscribed  circle."  The  problem  was  solved  by  Ko- 
bayashi  Koshin  in  1795,  and  the  relation  was  established  that 


where  i  =  the   diameter   of  the   inscribed    circle,  c  =  the  dia- 
meter of  the  circumscribed  circle,  and  /  =  the  given  product.1 
A  third  problem  is  as  follows:  "There  is  an  ellipse  in  which 
five    circles    are   inscribed   as  here  shown.     The  two  axes  of 


the  ellipse  being  a  and  b  it  is  required  to  find  the  diameter 
of  the  circle  A."  The  solution  as  given  by  Sano  Anko  in 
1787  may  be  expressed  as  follows: 


Another  problem  of  similar  nature  is  shown  in  Fig.  41,  from 
the  Zoku  Shimpeki  Sampo  (1806). 

A  style  of  problem  somewhat  similar  to  one  already  mention- 
ed in  connection  with  Arima  was  studied  in  1789  by  Hata 

1  For  the  case  of  a  square  of  side  2  we  have  2  J/l6=  8. 


IX.   The  eighteenth  Century. 


1 87 


Fig.  41.     From  the  Zoku  Shimpeki  Sampo  (1806). 

Judo,  as  follows:  "There  is  a  sphere  in  which  are  inscribed,  as 
in  the  figure,  two  spheres  A,  two  B,  and  two  C,  touching  each 


other   as  shown.     Given   the  diameters  of  A  and  C,  required 
to  find  the  diameter  of  B."     The  solution  given  is 


1 88  IX.  The  eighteenth  Century. 

Contemporary  with  Fujita  Sadasuke  was  Aida  Ammei  (1747— 
1817),  who  was  born  at  Mogami,  in  north-eastern  Japan.  Like 
Seki,  Aida  early  showed  his  genius  for  mathematics,  and  while 
still  young  he  went  to  Yedo  where  he  studied  under  a  certain 
Okazaki,  a  disciple  of  the  Nakanishi  school,  and  also  under 
Honda  Rimei,  although  he  used  later  to  boast  that  he  was  a 
self-made  mathematician,  and  to  assume  a  certain  conceit  that 
hardly  became  the  scholar.  Nevertheless  his  ability  was  such 
and  his  manner  to  his  pupils  was  so  kind  that  he  attracted 
to  himself  a  large  following,  and  his  school,  to  which  he  gave 
the  boastful  title  of  Superior  School,  became  the  most  popular 
that  Japan  had  seen,  save  only  Seki's.  Aida  wrote,  so  his 
pupils  say,  about  a  thousand  pamphlets  on  mathematics, 
although  only  a  relatively  small  number  of  his  contributions 
are  now  extant.  He  died  in  1817  at  the  age  of  seventy  years.1 

One  of  Aida's  works,  the  Tosei  Jinkoki  (1784)  deserves 
special  mention  for  its  educational  significance.  In  this  he 
discarded  the  inherited  problems  to  a  large  extent  and  sub- 
stituted for  them  genuine  applications  to  daily  life.  The  result 
was  a  great  awakening  of  interest  in  the  teaching  of  mathe- 
matics, and  the  work  itself  was  very  successful. 

Soon  after  the  publication  of  this  work  there  arose  an  un- 
fortunate controversy  between  Aida  and  Fujita  Sadasuke,  at 
that  time  head  of  the  Seki  School.  The  story  goes2  that 
Aida  had  at  one  time  asked  to  be  admitted  to  this  school, 
but  that  Fujita  in  an  imperious  fashion  had  told  him  that  first 
he  must  make  haste  to  correct  an  error  in  his  solution  of  a 
problem  that  he  had  hung  in  the  Shinto  shrine  on  Atago  hill 
in  Shiba,  Tokyo.  Aida  promptly  declined  to  change  his  solution 
and  thus  cut  himself  off  from  the  advantages  of  study  in  the 
Seki  school.  While  Aida  admits  having  visited  Fujita  he  says 
that  he  did  so  only  to  test  the  latter's  ability,  not  for  the 
purpose  of  entering  the  school. 

1  As    stated  upon    his    monument.     See  also   C.  KAWAKITA  in  the  Honcho 
Sugaku  Koenshu,   1908. 

2  This  account  is  digested    from    the   works    of  various  writers  who  were 
drawn  into  the  controversy. 


IX.  The  eighteenth  Century.  189 

As  a  result  of  all  this  unhappy  discussion  Aida  was  much 
embittered  against  the  Seki  school,  and  in  particular  he  set 
about  to  attack  the  Seiyo  Sampo  which  Fujita  Sadasuke  had 
published  in  1779.  For  this  purpose  he  wrote  the  Kaisei 
Sampo,  or  Improved  Seiyo  Sampo,  and  published  it  in  1785, 
criticising  severely  some  thirteen  of  Fujita' s  problems,  and  starting 
a  controversy  that  did  not  die  for  a  score  of  years.  Fujita's 
pupil,  Kamiya  Kokichi  Teirei,  then  wrote  in  the  former's  defence 
the  Kaisei  Sampo  Seiron,  and  sent  the  manuscript  to  Aida,  to 
which  the  latter  replied  in  his  Kaisei  Sampo  Kaisei-ron  which 
appeared  in  1786.  Kamiya  having  been  forbidden  by  Fujita 
to  publish  his  manuscript,  so  the  story  runs,  he  prepared 
another  essay,  the  Hi-kaisei  Sampo  which  also  appeared  about 
the  same  time,  the  exact  date  being  a  subject  of  dispute.  Of 
the  replies  and  counter-replies  it  is  not  necessary  to  speak  at 
length,  since  for  our  purposes  it  suffices  to  record  this  Newton- 
Leibnitz  quarrel  in  miniature.1  It  was  in  one  sense  what  is 
called  in  English  a  "tempest  in  a  tea-pot";  but  in  another  sense 
it  was  more  than  that,  for  it  was  a  protest  against  the  claims 
of  the  Seki  school,  of  the  individual  against  the  strongly 
entrenched  guild,  of  genius  against  authority,  of  struggling 


1  For  purposes  of  reference  the  following  books  on  the  controversy  are 
mentioned:  Fujita  wrote  a  reply  to  Aida  in  1786,  which  was  never  printed. 
Aida  wrote  the  Kaiwaku  Sampo  in  1788,  replying  to  the  Hi-kaisei  Sampo. 
Fujita  wrote  a  rejoinder,  the  Hi-kaiwaku  Sampo,  but  it  was  never  printed. 
Kamiya  published  the  Kaiwaku  Bengo  in  1789,  replying  to  Aida.  In  1792 
Aida  wrote  the  Shimpeki  Shinjutsu  in  which  he  criticised  the  Shimpeki  Sampo 
of  Fujita's  son,  and  also  wrote  the  Kaisei  Sampo  Jensho  in  which  he  criticised 
Fujita's  Seiyo  Sampo,  but  neither  of  these  was  printed.  In  1795  he  wrote 
his  Sampo  Kakujo,  an  abusive  reply  to  Kamiya,  but  in  the  same  year  he  wrote 
the  Sampo  Kokon  Tsiiran  (General  view  of  mathematical  works,  ancient  and 
modern)  in  which  he  has  something  good  to  say  of  him.  In  1799  Kamiya 
wrote  an  abusive  reply  to  Aida,  the  Hatsiiran  Sampo.  The  last  of  the  published 
works  by  the  contestants  was  Aida's  ffi-ffatsuran  Sampo  of  1801,  although 
the  controversy  still  went  on  in  unpublished  manuscripts.  The  manuscripts 
include  Kamiya's  Fukitsei  Sampo  (1803)  and  Aida's  Sampo  Senri  Dokko  (1804). 
Mention  should  also  be  made  of  the  Sampo  Tensho  ho  Shinan  (1811)  written 
by  Aida,  of  which  only  the  first  part  (5  books)  was  printed. 


IQO  IX.  The  eighteenth  Century. 

youth  against  vested  interests;  it  was  the  cry  of  the  insurgent 
who  would  not  be  downed  by  the  abuse  of  a  Kamiya  who 
championed  the  cause  of  a  decadent  monopoly  of  mathe- 
matical learning  and  teaching.  It  was  this  that  inspired  Aida 
to  act,  and  of  the  dignity  of  his  action  these  words,  from  a 
preface  to  one  of  his  works,  will  bear  witness:  "The  Seiyo 
Sampo*  treats  of  subjects  not  previously  worked  out,  and 
certain  of  its  methods  have  never  been  surpassed.  The  author's 
skill  in  mathematics  may  safely  be  described  as  unequalled  in 
all  the  Empire.  Upon  this  work  the  student  may  in  general 
rely,  although  it  is  not  wholly  free  from  faults.  Since  it  would 
be  a  cause  of  regret,  however,  if  posterity  should  be  led  into 
error  through  these  faults,  as  would  be  the  natural  influence 
of  so  great  a  master  as  Fujita,  I  have  taken  the  trouble  to 
compose  a  work  which  I  now  venture  to  offer  to  the  world 
as  a  guide."  Such  words  and  others  in  recognition  of  Fujita's 
merits  did  not  warrant  the  abuse  that  Kamiya  heaped  upon 
Aida,  and  the  impression  left  upon  the  reader  of  a  century 
later  is  that  of  a  staunch  champion  of  liberty  of  thought,  corn- 
batted  by  the  unprovoked  insults  and  unjust  scorn  of  vested 
interests.  Fujita  seems  to  have  solved  his  problems  correctly 
but  to  have  expressed  his  work  in  cumbersome  notation,2 
while  Aida  stood  for  simplicity  of  expression.  Neither  was 
in  general  right  in  attacking  the  solutions  of  the  other,  and  in 
the  heat  of  controversy  each  was  led  to  statements  that  were 
incorrect.  The  whole  struggle  is  a  rather  sad  commentary  on 
the  state  of  mathematics  in  the  waning  days  of  the  Seki  school, 
when  the  trivial  was  magnified  and  the  large  questions  of 
mathematics  were  forced  into  the  background. 

Aida  was  an  indefatigable  worker,  practically  his  whole  life 
having  been  spent  in  study.  As  a  result  he  left  hundreds  of 
manuscripts,  most  of  which  suffered  the  fate  of  so  many 


1  Fujita's  work  of  1779. 

2  As  compared   with    that   of  Aida,   although    an    improvement   upon  that 
of  his  predecessors. 


IX.  The  eighteenth  Century.  19 1 

thousands  of  books  in  Japan,  the  fate  of  destruction  by  fire.1 
Of  the  contents  of  the  Sampo  Kokon  Tsiiran  (1795)  already 
mentioned,  only  a  brief  note  need  be  given.  In  Book  VI  Aida 

gives  the  value  of  --  as  follows: 

2!  3!  4! 


He  gives  a  series  for  the  length  of  an  arc  x  in  terms  of  the 
chord  c  and  height  //  thus: 

2  2.4  2.4.6  . 
x  =  c  (i  -\ m  H ;;/  +  -      —m  +•••), 

3  3-5  3-5-7 

where       m  =  — = ^ 


and  ^  is  the  diameter  of  the  circle.  In  the  same  work  he 
gives  a  formula  for  the  area  of  a  circular  segment  of  one 
base: 

he    .  2  2.4  2.4.6  , 


Aida    also    gave   a  solution  of  a  problem  found  in  Ajima's 
Fnkyu  Sampd,  as  follows:  The  side  of  an  equilateral  triangle 
is  given  as   an  integer  n.     It  is 
required    to    draw  the  lines  slf 
s2,   .  .  .,    parallel    to    one    side, 
such  that  the  /'s,  g's  and  s's 
as    shown    in    the    figure    shall 
all  have  integral  values. 

Ajima  had  already  solved  this 
before  Aida  tried  it,  and  this 
is,  in  substance,  his  solution: 
Decompose  n  into  two  factors,  n=ab 

a     and     b,     which    are     either 

both  odd  or  both  even.  If  this  cannot  be  done  a  solution  is 
impossible.  The  rules  are  now,  as  expressed  in  formulas,  as 
follows  : 


KAWAKITA'S  article  in  the  Honcho  Siigaku  Koenshu,  p.  13. 


IX.  The  eighteenth  Century. 

/!  =  k2  —a2,  ql  =  (k  —  a)2  —  ka, 

pz  =  A  —  D,  p$  =  sz—  D,  .  .  . 

s2  =  n  —  -/!,  -$-3  =  J2  —  /2,  •  •  • 


where  k  =  ±  (a  +  b\      D=^(b-a}2,      M=^D. 

When  /i  >  —  w  it  may  be  taken  at  once  for  s2  and  n  —  s2 
for  /x.* 

Aida  objects  to  the  length  of  such  a  rule,  and  he  proposes 
to  solve  the  problem  thus: 

Let  n  =  ab,  where  a  <  b. 
Then  let  -i-  (&  —  a)  =  D. 

Then  («  —  Z?)  (b-D)=s2, 


Also  let  sr 
and  we  have 


Aida  also  did  some  work   in   indeterminate  equations2  and 
was  the  first   to  take  up  the  permutation  of  magic  squares.  3 


1  Ajima  does  not  tell  what  to  do  for  q-i  if  \k  —  a)2  <  ka. 

2  As   in   solving   2*  =  x-i2  -\-x22  +  ^3*  +  -*"42  +  ^52.      See    the    article    by 
C.  HITOMI   in   the  'Journal  of  the  Tokyo  Physics   School.     From  Aida's  manu- 
script Sampo  Seisii-jutsu    (On   the   method    of  solutions   in   integers),   we  also 
take  the  following  types: 

I2  ^I2  +  22  X22   +   32  A'f  -\ f-   TO2  ^2IO  ==^2 

and 

I*!2    +    2JC22   +3^32    + 1-    IO.*lio=-J/2. 

This  manuscript  was  probably  written  not  earlier  than  1807. 

3  Upon   the    authority    of   K.   KANO,    to   whom    we    are   indebted   for  the 
statement. 


IX.  The  eighteenth  Century.  193 

He  also  gives  an  ingenious  method  for  expanding  a  binomial, 

or  rather  for  writing    down   the  coefficients  in  the  expansion 

i 

of  (a  +  b}n  ,  which  expresses  roots  in  series. 

One  of  the  most  interesting  of  Aida's  solutions  is  that  of 
the  problem  to  find  the  radius  r  of  a  regular  w-gon  of  side  s.1 
He  says  that  of  the  infinite  series  representing  —  the  successive 
terms  are 


4.6  '  4-6.8.10 

3--  (4)']  [5-       1 

4.6.8.  10.12.14 
If  we  put  m  for  —  ,  and  x  for  —  ,  the  series  becomes 

n.'  2    ' 

sin  (m  arc  sin  x)  m          m  (m*  —  i2)       2 

~^~~  "  H  "  3  ! 

m  (m2  _  T2)  (m,  —  32) 

5-  '  *' 

a  series  that  has  been  attributed  both  to  Newton  and  to  Euler. 
We  therefore  have 


6 
=  2  sin  I  —  arc  sin 


.       l  \ 
in  —  )  , 

2  J  ' 


S  .         IT 

or  —  =  2  sin  —  , 


whence  sin  -r-  =  —  .    It  is  generally  conceded  that  Aida  knew 

that   the   formula    had    already   been   given   in    substance    by 
Kurushima.2     It  also  appeared  in  Matsunaga's  Hoyen   Sankyo 

of  1739- 

From  the  names  considered  in  this  chapter  we  might  charac- 
terize the  eighteenth  century  as  one  of  problem-  solving,  of  the 
extension  of  a  rather  ill-defined  application  of  infinite  series 

1  HAYASHI,  History,  part  II,  p.  13. 

2  See  p.  176. 


IX.    The  eighteenth  Century. 

to  the  mensuration  of  the  circle,  of  some  slight  improvement 
in  the  various  processes,  of  the  rather  arrogant  supremacy  of 
the  Seki  school,  and  of  a  bitter  feud  between  the  independents 
and  the  conservatives  in  the  teaching  of  mathematics.  And 
this  is  a  fair  characterization  of  most  of  the  latter  half  of  the 
century.  There  was,  however,  one  redeeming  feature,  and  this 
is  found  in  the  work  of  Ajima  Chokuyen,  of  whom  we  shall 
speak  in  the  next  chapter. 


CHAPTER  X. 
Ajima  Chokuyen. 

In  the  midst  of  the  unseemly  strife  that  waged  between 
Fujita  and  Aida  in  the  closing  years  of  the  eighteenth  century 
there  dwelt  in  peaceful  seclusion  in  Yedo  a  mathematician  who 
surpassed  both  of  these  contestants,  and  who  did  much  to 
redeem  the  scientific  reputation  of  the  Japanese  of  his  period. 
A  man  of  rare  modesty,  content  with  little,  taking  delight  in 
the  simple  life  of  a  scholar  rather  than  in  the  attractions  of 
office  or  society,  almost  unknown  in  the  midst  of  the  turmoil 
of  the  scholastic  strife  of  his  day,  Ajima  Manzo  Chokuyen  * 
was  nevertheless  a  rare  genius,  doing  more  for  mathematics 
than  any  of  his  contemporaries. 

He  was  born  in  Yedo  in  1739,  and  as  a  samurai  he  served 
there  under  the  Lord  of  Shinjo,  whose  estates  were  in  the 
north-eastern  districts.  He  was  initiated  into  the  secrets  of 
mathematics  by  one  Iriye  Ochu2,  who  had  studied  in  the 
school  of  Nakanishi.  He  afterwards  became  a  pupil  of  Yamaji 
Shuju,  and  at  this  time  he  came  to  know  Fujita  Sadasuke  with 
whom  he  formed  a  close  friendship  but  with  whose  controversy 
with  Aida  he  never  concerned  himself.  And  so  he  received 
a  training  that  enabled  him  to  surpass  all  his  fellows  in  solving 
the  array  of  problems  that  had  accumulated  during  the  century, 
including  all  those  which  had  long  been  looked  upon  as  wholly 
insoluble.  Such  a  type  of  mind  rarely  extends  the  boundaries 
of  mathematical  discovery,  but  occasionally  an  individual  is 

1  See  also  HARZER,  P ,  loc.  cit.,  p.  34  of  the  Kiel  reprint  of  1905. 

2  Also  given  as  Irie  Masatada. 

13* 


196  X.  Ajima  Chokuyen. 

found  with  this  kind  of  genius  who  is  at  least  able  to  help  in 
improving  science  by  his  genuine  sympathy  if  not  by  his 
imagination.  Such  a  man  was  Ajima.  His  interests  extended 
from  tenzan  algebra  to  the  Diophantine  analysis,  and  from 
simple  trigonometry  to  a  new  phase  of  the  yenri  which  had 
occupied  so  much  attention  throughout  the  century.  Possessed 
of  the  genius  of  simplicity,  he  clothed  in  more  intelligible  form 
the  abstract  work  of  his  predecessors,  even  if  he  made  no 
noteworthy  discovery  for  himself.  Although  his  retiring  nature 
would  not  allow  him  to  publish  his  works,  he  left  many  manu- 
scripts of  which  the  more  important  may  well  occupy  our 
attention.  He  died  in  1798  at  the  age  of  fifty-nine  years,1 
honored  by  his  fellows  as  a  Meijin2  (genius,  or  person  dexterous 
in  his  art)  in  the  field  in  which  he  labored. 

In  the  Kan-yen  Muyuki*  (1782)  he  gives  a  solution  in  integers 
of  the  problem  of  n  tangent  circles  described  within  a  given 
circle,  and  similarly  for  an  array  of  circles  tangent  to  one 
another  and  to  the  -given  circle  externally.  The  problem  is 
one  of  those  in  indeterminate  analysis  to  which  the  Japanese 
scholars  paid  much  attention.  Another  indeterminate  equation 
considered  by  him  is  the  following: 

xS  +  xS  +  xj  +  x*  +  x52  =  j2. 

This  appears  in  a  manuscript  entitled  Beki-wa  Kaiho  Mu-yuki 
Seisu-jutsu  (Integral  solutions  for  the  square  root  of  the  sum 
of  squares)  and  dated  1791. 

Another  work  of  his  was  the  Sampo  Kosofi  in  which  the 
famous  Malfatti  problem  appears,  to  inscribe  three  circles  in  a 
triangle,  each  tangent  to  the  other  two.  Ajima  does  not, 
however,  consider  the  geometric  construction,  preferring  to 
attack  the  question  from  the  standpoint  of  algebra,  after  the 
usual  manner  of  the  Japanese  scholars.  The  problem  first 

1  C.  KAWAKITA,   in   his   article   in   the    Honcho  Sugaku  Koenshu   says   that 
he  is  sometimes  thought  to  have   died  in   1800,   but  the  date  given  by  us  is 
from  the  records  of  the  Buddhist  temple  where  he  is  buried. 

2  The  term  may  be  compared  to  pandit  in  India. 

3  Literally,  Integral  solutions  of  circles  touching  a  circle. 

4  Literally,  A  draft  of  a  mathematical  problem. 


X.  Ajima  Chokuyen.  197 

appears  in  Japan,  so  far  as  now  known,  in  the  Sampo  Gakkai* 
published  by  Ban  Seiyei  of  Osaka  in  1781,  the  solution  being 
much  more  complicated  than  that  given  subsequently  by 
Ajima.2 

The  Senjo  Ruiyen-jutsu*  and  the  Yennai  Yo-ruiyen-jutsu* 
are  two  works  upon  groups  of  circles  tangent  to  a  straight 
line  and  a  circle,  or  to  two  circles.  In  the  Renjutsu  Henkan 
(1784)5  he  treats  the  subject  still  more  generally,  considering 
the  straight  line  as  a  limiting  case  of  a  circumference. 

The  Jnji-kan  Shinjutsuf  a  manuscript  of  1794,  considers  the 
question  of  an  anchor-ring  cut  by  two  cylinders,  a  problem 
first  studied  in  Japan  by  Seki,  and  later  by  Arima  in  his  Shuki 
Sampo  (1769),  where  infinitesimal  analysis  seems  to  have  been 
applied  to  it  for  the  first  time  in  this  country.  One  of  the 
most  famous  problems  solved  by  Ajima  is  that  known  as  the 
Gion  Temple  Problem,  and  treated  by  him  in  his  Gion  Sandai 
no  KaiJ  The  problem  is  as  follows:  "There  is  a  segment  of 
a  circle,  and  in  this  there  are  inscribed,  on  opposite  sides  of 
the  altitude,  a  circle  and  a  square.  Given  the  sum  of  the 
chord,  the  altitude,  the  diameter  of  the  inscribed  circle,  and  a 


1  Literally,  Sea  of  learning  for  mathematical  methods. 

2  ENDO,  Book  III,  p.  187.     For  the   history  of  the  problem   in   the  West 
see    A.    WlTTSTElN,    Geschichte    des   Malfatti' schen    Problems,    Miinchen,     18(7, 
Diss. ;    M.   BAKER   in    the  Bulletin  of  the  Philosophical  Society  of  Washington, 
Vol.  IT,  p.  113;    Intorno  alia  vita  ed  agli  scritti  di   Gianfranco  Malfatti,    in  the 
Boncompagni    Bulletino,    tomo    IX,    p.  361.      For   the   isosceles    triangle    the 
problem    appears   in    the    Opera   of  Jakob  Bernoulli,  Geneva,   1744,   Problema 
geometrica,    lemma  II,    tomus  I,   p.  303.     It   was    first   published   by   Malfatti 
(1731 — 1807)  in  the  Memorie  di  Matematica  e  di  Fisica,  Modena,  1803,  tomo  X, 
p.  235,  five  years  after  Ajima  died. 

3  Literally,   On    Circles  described  successively   on   a  line.     It   appeared  in 
1784,  and  a  sequel  1791. 

4  Literally,  On  Circles  described  successively  within  a  circle. 

5  Literally,  The  Adapting  of  a  general  plan  to  special  cases. 

6  Literally,  Exact  method  for  the  cross-ring. 

7  Literally,  The   Analysis   of  the   Gion  Temple  problem.     The  manuscript 
is    dated   the   24^  day    of    the    6''>  month,    1773,    although   ENDO   (Book  III, 
p.  8)  gives    1774  as  the  year. 


X.  Ajima  Chokuyen. 

side   of  the  square,  and    also  given  the  sum  of  the  quotients 
of  the  altitude  by  the  chord,  of  the  diameter  of  the  circle  by 


the  altitude,  and  of  the  side  of  the  square  by  the  diameter 
of  the  circle,  it  is  required  to  find  the  various  quantities 
mentioned." 

The  problem  derives  its  name  from  the  fact  that  it  was,  with 
its  solution,  first  hung  before  the  Gion  Temple  in  Kyoto  by 
Tsuda  Yenkyu,  a  pupil  of  Nishimura  Yenri's1,  the  solution 
depending  upon  an  equation  of  the  1024^  degree  in  terms  of 
the  chord.  The  solution  was  afterward  simplified  by  one 
Nakata  so  as  to  depend  upon  an  equation  of  the  46*  degree. 
Ajima  attacked  the  problem  in  the  year  1774,  and  brought  it 
down  to  the  solution  of  an  equation  of  the  10*  degree.  This 
is  not  only  a  striking  proof  of  Ajima's  powers  of  simplification, 
but  it  is  also  evidence  of  the  improvement  constantly  going 
on  in  the  details  of  Japanese  mathematics  in  the  eighteenth 
century. 

Ajima  considers  in  his  Fujin  Isshu  (Periods  of  decimal 
fractions)  the  problem  of  finding  the  number  of  figures  con- 
tained in  the  repetend  of  a  circulating  decimal  when  unity  is 
divided  by  a  given  prime  number.  Although  he  states  that 
the  problem  is  so  difficult  as  to  admit  of  no  general  formula, 
he  shows  great  skill  in  the  treatment  of  special  cases.  To 
assist  him  he  had  the  work  of  at  least  two  predecessors,  for 
Nakane  Genjun  had  studied  the  problem  for  special  cases  in 
his  Kanto  SampO  of  1738,  and  in  the  Nisei  Hyosen  Ban  Seiyei 
of  Osaka  had  given  the  result  for  a  special  case,  but  without 

1  Whose  Tengaku  Shiyo  (Astronomy  extract)  was  published  in  1/76. 


X.  Ajima  Chokuyen.  199 

the  solution.  Ajima  was,  however,  the  first  Japanese  scholar 
to  consider  it  in  a  general  way. 

He  first  gives  a  list  of  numbers  from  which,  considered  as 
divisors  of  unity,  there  arise  periods  of  from  I  to  16  figures, 
as  follows: 

1  figure  3 

2  figures  1 1 

3  figures  37 

4  figures  101 

5  figures  41,  271 

6  figures  7,  13 

7  figures  239,  4649 

8  figures  73  137 

9  figures  333,667 

10  figures  9091 

11  figures  21,649,  5 i  3,239 

12  figures  9901 

13  figures  53,  79,                            665,371,653 

14  figures  909,091 

15  figures  31,  2,906,161 

16  figures   17,  5,882,353. 

As  an  example  of  his  methods  we  will  consider  his  treat- 
ment of  the  special  fractions  and  .  Ajima  assumes 

353  103 

without  explanation  that  the  required  numbers  are  given  by 
one  of  the  possible  products  of  some  of  the  prime  factors  in 

353  -  i  =  352  =  25xii 
and  103  —  i  =  102  =  2x3x17, 

respectively.  He  then  says  that  out  of  these  products  it  can 
be  found  by  trial  that  the  respective  numbers  sought  are  32 
and  34,  but  he  does  not  tell  how  this  trial  is  effected.  This 
was  done  later  by  Koide  Shuki  (1797 — 1865)  and  the  result 
appeared  in  print  in  the  Sampo  Tametebako  (1879),  a  work 
by  Koide's  pupil,  Fukuda  Sen,  who  wrote  under  the  nom  de 


2OO  X.   Ajima  Chokuyen. 

plume   Riken.      Koide    merely    explains    Ajima's    work,    using 
identically  the  same  numbers. 

Neither  his  explanation  nor  Ajima's  hint  is,  however,  very 
clear,  and  each  shows  both  the  difficulties  met  by  followers 
of  the  wasan  and  their  tendency  to  keep  such  knowledge  from 
profane  minds. 

r  n 

For  the  expansion  of  V~N  Ajima  gives  two  formulas,1  which 
may  be  expressed  in  modern  notation  as  follows: 


—       T~>  ,          —      „ 

=a^  --  am  --    —D*m  +  -         D2m  — 

n  2«  3« 


3  «  — 


r      i       ^  <    N'+I  (»—IH«—  2)  •••[(»—  i)  «—  ii    2-i 

#  n  --  m  +>(—i)  —  r-1-        L  /«'  , 

»  2  «  •  2  tt  •  3  «  •  •  •  •  «  « 


where  m  = 


-*-1  «  •  2  w  •  3  n  •  •  •  i  n 

where  m  =  — -,"— .     No   explanation   of  the   work    is    given. 

He  also  treated  of  square  roots  by  means  of  continued  fractions, 
the  convergents  of  which  he  could  obtain.2 

Ajima  also  studied  the  spiral  of  Archimedes,  although  not 
under  that  name.^  It  had  been  considered  even  before  Seki's 
time,4  and  Seki  himself  gave  some  attention  to  it.s  Lord  Arima 
also  discussed  it  in  his  S/iuki  Sampo  of  1769.  It  is  to  Ajirna, 
however,  that  we  are  indebted  for  the  only  serious  treatment 
up  to  his  time.  He  divided  a  sector  of  a  circle  by  radii  into 
n  equal  parts,  and  then  divided  each  of  the  radii  also  into  n 
equal  parts  by  arcs  of  concentric  circles.  He  then  joined 
successive  points  of  intersection,  beginning  at  the  center  and 

1  In  the  Tetsu-jutsu  Kappo  of  1784. 

2  HAYASHT,   History,   part  II,   p.  9,   probably  refers  to  his   commentary  on 
Kurushima's  method. 

3  It  was  called  by  Japanese  scholars  yenkei,  yempai,  or  yen-wan. 

4  As  in  Isomura's  Ketsugisho  of  1684. 

5  In  his  Kai-Kendai  no  Ho,  and  reproduced  in  the  Taisei  Sankyo. 


X.  Ajima  Chokuyen.  2OI 

ending  on  the  outer  circle,  and  said  that  the  limiting  form  of 
this  broken  line  for  ;/  =  oo  was  the  yempai.  He  then  preceded 
to  find  the  area  between  the  curve  and  the  original  arc  by 
finding  the  trianguloid  areas  and  summing  these  for  n  =  °o, 

obtaining  —  ar.  In  a  similar  fashion  he  rectifies  the  curve, 
obtaining  as  a  result  the  series 

s  =  r    ,     a*_  _    a<      ,        a6  , 


6r         407-3  H2;-5          11527-7  28167-9 

a  result   that   Shiraishi    Ch5chu    (1822)    puts    in  a  form  equi- 
valent to 


Ajima  also  gives  a  formula  for  the  square  of  the  length  of  the 
curve,  and  summarizes  his  work  by  giving  numerical  values 
for  r  =  10,  a  =  5,  thus: 

s  =  10.402288144  .  .  . 
s2=  108.2075996685  ..., 

from  which  he  concludes  that  Seki's  treatment  of  the  subject 
was  rather  crude. 

Ajima  made  a  noteworthy  change  in  the  yenri,  in  that  he 
took  equal  divisions  of  the  chord  instead  of  the  arc,  thus 
simplifying  the  work  materially.1  Indeed  we  may  say  that  in 
this  work  Ajima  shows  the  first  real  approach  to  a  mastery 
of  the  idea  of  the  integral  calculus  that  is  found  in  Japan, 
which  approach  we  may  put  at  about  the  year  1775.  Since 
this  work  was  so  noteworthy  we  enter  upon  a  more  detailed 
description  than  is  usually  required  in  speaking  of  the  achieve- 
ments of  the  eighteenth  century. 

Ajima  proceeds  first  to  find  the  area  of  a  segment  of  a 
circle  bounded  by  two  parallel  lines  and  the  equal  arcs  inter- 

i  This  appears  in  his  Kohai-jutsu  Kai  (Note  on  the  measurement  of  an 
arc  of  a  circle),  the  date  of  which  is  not  known.  ENDO  (Book  III,  p.  l) 
thinks  that  it  precedes  his  knowledge  of  the  yenri  as  imparted  by  his 
teacher  Yamaji. 


202 


X.  Ajima  Chokuyen. 


cepted  by  them,  that  is,  the  area  ABCD  in  the  figure.    Here 
we  divide  the  chord  c  of  the  arc  into  n  equal  parts.1 


Then  from  the  figure  it  is  apparent  that 


where  pr  is  the  rth  parallel  from  the  diameter  d. 

Ajima  now  expands  pr,  without  explaining  his  process  (evi- 
dently that  of  the  tetsujutsii),  and  obtains 

3         / 


,r       i/rjiv     _i 

"TV  rf  )  ~~  8 


_3 

&  \~4 


384  V  rf 


] 

1 

" 


i  In   the   figure   the   chord  DC  is   divided   into    5    equal   parts,    each  part 
being  designated  by  fi,  so  that  5ja  =  <r. 


X.  Ajima  Chokuyen.  203 

Summing  for  r  =  I,  2,  3  •  •  •  n,  and  multiplying  by  u.  we  have 
the  following  series: 


-- 

48     d  384 


.  (2«3 


5  «4  + 

2I«S  — 


66  #s  _ 


-  50027/9  +  8 5 80 #7  —  90097/5  +  45507/3  —  691  n) 
1 

Now    substituting    for    |i    its    value,   — ,    and    then    letting   n 

approach  =»,  all  terms  with  n  in  the  denominator  approach  o 
as  a  limit,  and  the  limit  to  which  the  required  area  ap- 
proaches is 

j  f  I        r3  I        <r5  3         c1 

area  =  a  \  c  — • ^  •  - 

6      </2        40     </+        336     d*> 
15        c9          105         r»  945         ri3 


3456      ^8        43240       </i°          599040 


204  X.  Ajima  Chokuyen. 

From  this  Ajima  easily  derives  the  area  of  the  segment,  and 
from  that  he  gets  the  length  of  the  arc,  as  follows: 

_  r  +     l2      ^  4.      12'32     .  £   ,         i*-3'-2 
" 


2-3      <*'       a  «  3  •  4  •  f  2.3.4.5.6.7      ^6 

+    •  •  -, 

from  which  other  formulas  may  be  derived. 

Ajima  also  directed  his  attention  to  the  problem  of  rinding 
the  volume  cut  from  a  cylinder  by  another  cylinder  which 
intersects  it  at  right  angles.  His  result  is  given  by  his  pupil 
Kusaka  Sei  (1764  —  1839)1  in  his  manuscript  work,  the  Fukyu 
Sampo  (1799),  without  explanation,  as  follows: 

kz  d 


•  ^-  \  i 
4   I 

—  •  •  •  \ 


}'  '      ' 


—  -o-  ..  0   o  — 

8  -8-  16-  16  8  -8-  16-  16  -40 

where  k  and  d  are  the  diameters  of  the  pierced  and  piercing 
cylinders,  respectively,  and  where  m  =  k2  -'.-  d2.  2  In  another 
work  of  I794,3  however,  Ajima  gives  an  analysis  of  the  problem, 
cutting  the  solid  into  elements  as  in  the  case  of  the  segment 
of  a  circle  already  described.  He  then  proceeds  to  the  limit 
as  in  that  case,  and  thus  gives  a  good  illustration  of  a  fairly 
well  developed  integral  calculus  applied  to  the  finding  of 
volumes.4 

Thus  we  at  last  find,  in  Ajima's  work,  the  calculus  established 
in  the  native  Japanese  mathematics,  although  possibly  with 
considerable  European  influence.  With  him  the  use  of  the 
double  series  again  appears,  it  having  already  been  employed 
by  Matsunaga  and  Kurushima,  and  by  him  the  significance 
of  double  integration  seems  first  to  have  been  realized.  He 


1  Or  Kusaka  Makoto. 

2  ENDO  attempts  some  explanation  in  his  History,  Book  III,  p.  25. 

3  This    is   a   manuscript   of   the   Yenchii   Sen-kiiyen  Jutsti  (Evaluation   of  a 
cylinder  pierced  by  another). 

4  The  work  as  given  by  Ajima  is  too  extended  to  be  set  forth  at  length, 
the  theory  being  analogous  to  that  which  has  already  been  illustrated. 


X.  Ajima  Chokuyen.  205 

lacked  the  simple  symbolism  of  the  West,  but  he  had  the 
spirit  of  the  theory,  and  although  his  contemporaries  failed 
to  realize  his  genius  in  this  respect,  it  is  now  possible  to 
look  back  upon  his  work,  and  to  evaluate  it  properly.  As 
a  result  it  is  safe  to  say  that  Ajima  brought  mathematics  to 
a  higher  plane  than  any  other  man  in  Japan  in  the  eighteenth 
century,  and  that  had  he  lived  where  he  could  easily  have 
come  into  touch  with  contemporary  mathematical  thought  in 
other  parts  of  the  world  he  might  have  made  discoveries  that 
would  have  been  of  far  reaching  importance  in  the  science. 


CHAPTER  XI. 
The  Opening  of  the  Nineteenth  Century. 

The  nineteenth  century  opened  in  Japan  with  one  noteworthy 
undertaking,  the  great  survey  of  the  whole  Empire.  At  the 
head  of  this  work  was  Ino  Chukei,1  a  man  of  high  ability  in 
his  line,  and  one  whose  maps  are  justly  esteemed  by  all  cartog- 
raphers. Until  he  was  fifty  years  of  age  he  lived  the  life  of 
a  prosperous  farmer.  While  not  himself  a  contributor  to  pure 
mathematics,  he  came  in  later  life  under  the  influence  of  the 
astronomer  Takahashi  Shiji2  (1765  —  1804),  and  at  the  solicitation 
of  this  scholar  he  began  the  work  that  made  him  known  as 
the  greatest  surveyor  that  Japan  ever  produced.  Takahashi 
seems  to  have  become  acquainted  with  Western  astronomy 
and  spherical  trigonometry  through  his  knowledge  of  the  Dutch 
language.  He  had  also  studied  astronomy  while  serving  as  a 
young  man  in  the  artillery  corps  at  Osaka,  his  teacher  having 
been  a  private  astronomer  and  diligent  student  named  Asada 
Goryu  (1732—1799),  by  profession  a  physician.  This  Asada 
was  learned  in  the  Dutch  sciences^  and  is  sometimes  said  to 
have  invented  a  new  ellipsograph.4  In  1795  he  was  called  to 


1  Or  InO  Tadanori,  InO  Tatayoshi,  whose   life   and   works  are  now  (1913) 
being  studied  by  Mr.  R.  Otani. 

2  Or  Takahashi  Shigetoki,  Takahashi  Yoshitoki,  Takahashi  Munetoki. 

3  As  only  physicians  and  interpreters  were  at  this  time. 

4  A  different  instrument  was   invented  by  Aida  Ammei,  who  left  a  manu- 
script work   of  twenty   books    upon  the   ellipse.     There  is  also  a  manuscript 
written   by  Hazama  Jushin   in  1828,    entitled  Dayen  Kigen  (A  description  of 
the   ellipse)   in    which  it   is   claimed   that   the    ellipsograph   in   question    was 
invented    by    the    writer's    father,    Hazama    Jufu    (or    Shigetomi)    who    lived 


XI.   The  Opening  of  the  Nineteenth  Century.  207 

membership  in  the  Board  of  Astronomers  of  the  shogunate, 
an  honor  which  he  declined  in  favor  of  his  pupils  Takahashi 
Shiji  and  Hazama  Jufu.  Takahashi  thereupon  took  up  his 
residence  in  Yedo,  where  he  died  in  I8O4,1  five  years  after 
Asada  had  passed  away. 

Among  Asada's  younger  contemporaries  was  Furukawa  Uji- 
kiyo  (1758 — 1820),  who  founded  a  school  which  he  called  the 
Shisei  Sanka  Ryu.2  He  was  a  shogunate  samurai  of  high 
rank,  holding  the  office  of  financial  superintendent,  and  although 
a  prolific  writer  he  contributed  little  of  importance  to  mathe- 
matics. 3  Nevertheless  his  school  flourished,  although  it  was 
one  of  nineteen*  at  that  time  contending  for  mastery  in  Japan, 


from  1756  to  1816,  and  that  it  dated  from  the  beginning  of  the  Kwansei 
era  (1789 — 1800).  Hazama  Juffi  was  a  pupil  of  Asada's,  and  was  a  merchant. 

1  It  is  said  at  about  the  age  of  forty. 

z  School  of  Instruction  with  Greatest  Sincerity.  It  was  also  called  the 
Sanwa-itchi  school. 

3  His  Sanseki,  a  collection  of  tenzan  problems  consists  of  223  books. 

4  ENDO,  Book  III,  p.  57.     On  account  of  the  importance  of  these  schools 
in  the  history  of  education  in  Japan,  the  list  is  here  reproduced  for  Western 
readers : 

1.  Momokawa  Ryu,  or  Momokawa's  School,  teaching  the  soroban  arithmetic 
as  set  forth  in  Momokawa's  Kameizan  of  1645. 

2.  Seki  Ryu,  or  Seki's  School. 

3.  Kuichi  Ryu.     The  meaning  is  not  known. 

4.  Nakanishi  Ryu,  or  Nakanishi's  School. 

5.  Miyagi  Ryu,  or  Miyagi's  School. 

6.  Takuma  Ryfi,  or  Takuma's  School. 

7.  SaijO  Ryu,  or  Superior  School,  sometimes  incorrectly  given  as  Mogami 
School. 

8.  Shisei  Sanka  Ryu,  or  Sanwa    Itcjii  Ryu.     The   latter   name    may  mean 
the  Agreement  of  Trinity  School. 

9.  Koryu,  the 'Old  School;  or  Yoshida  Ryu,  Yoshida's  School. 

10.  Kurushima  Gaku,  or  Kurushima's  School. 

11.  Ohashi  Ryu,  or  Ohashi's  School. 

12.  Xakane  Ryu,  or  Nakane's  School,  the  Takebe-Nakane  sect  of  the  Seki 
School. 

13.  Nishikawa  Ryu,  or  Nishikawa's  School. 

14.  Asada  Ryu,  or  Asada's  School. 

15-  Hokken  Ryu.     The  meaning  is  not  known. 


2C>8  XI.  The  Opening  of  the  Nineteenth  Century. 

and  when  he  died  it  was  continued  by  his  son,  Furukawa  Ken 

(1783-1837). 

In  this  school,  as  in  others  of  its  kind,  the  tenzan  algebra 
attracted  much  attention.  It  will  be  recalled  that  it  was  first 
made  public  in  the  Shuki  Sampo,  composed  by  Arima  in  1769, 
a  treatise  written  in  Chinese  characters  and  in  such  an  obscure 
style  as  not  easily  to  be  understood.  No  better  treatment 
appeared,  however,  until  one  was  set  forth  by  Sakabe  Kohan 
(1759 — 1824)'  in  1810  under  the  title  Sampo  Tenzan  Shinan- 
Roku.2  In  the  same  year  two  other  works  were  written  upon 
this  subject,  one  by  Ohara  Rimei^  and  the  other  by  Aida,* 
but  neither  of  these  had  the  merit  of  Sakabe's  treatise.  Sakabe 
was  in  his  younger  days  in  the  Fire  Department  of  the  sho- 
gunate,  but  he  early  resigned  his  post  and  became  a  ronin  or 
free  samurai,  devoting  all  of  his  time  to  study  and  to  the 
teaching  of  his  pupils.  He  first  learned  mathematics  from 
Honda  Rimei  (1744 — 1821),  who  was  a  leader  of  the  Takebe- 
Nakane  sect  of  the  Seki  school,  a  man  who  was  more  of  a 
patriot  than  a  mathematician,  but  who  knew  something  of  the 
Dutch  language  and  who  was  the  first  Japanese  seriously  to 
study  the  science  of  navigation  from  European  sources.  Sakabe 
also  studied  in  the  Araki-Matsunaga  school  and  was  one  of 
the  most  distinguished  pupils  of  Ajima.  He  left  a  noble  record 
of  a  life  devoted  earnestly  to  the  advance  of  his  subject  and 
to  the  assistance  of  his  pupils. 


16.  Komura  Ryu,  or  Komura's  School,  a  school  of  surveying. 

17.  Furuichi  Ryfl,  or  Furuichi's  School. 

1 8.  Mizoguchi  Ryu,  or  Mizoguchi's  School,  a  school  of  surveying. 

19.  Shimizu  Ryu,  or  Shimizu's  School,  also  a  school  of  surveying. 

1  He   was    a  prolific   writer,    his    other   more   important  works   being^the 
Shinsen   Tetsujutsu   (1795)    and    the    Kakujntsu-keimo    (Considerations    on    the 
theory  of  the  polygon,   1802).     These  exist    only  in  manuscripts.     His  literal 
name  was  Chugaku. 

2  Exercise  book  on  the  tenzan  methods. 

3  Tenzan  Shinan  (Exercises  in    the    tenzan   method).     Ohara  died  in  1831. 

4  Sampo  Tensho-ho,   or  Sampo   Tensei-ho,   Treatise   on    the   Tensho   method. 
Aida  called  the  tenzan  method  by  the  name  tensho. 


XI.  The  Opening  of  the  Nineteenth  Century.  209 

Sakabe's  treatise  was  published  in  fifteen  Books,  the  last  one 
appearing  in  1815.  One  of  the  first  peculiarities  of  the  work 
that  strikes  the  reader  is  the  new  arrangement  of  the  sangi, 
which  it  will  be  recalled  were  differently  placed  for  alternate 
digits  by  all  early  writers.  Sakabe  remarks  that  "it  is  ancient 
usage  to  arrange  these  sometimes  horizontally  and  sometimes 
vertically,  .  .  .  but  this  is  far  from  being  a  praiseworthy  plan, 
it  being  a  tedious  matter  to  rearrange  whenever  the  places  of 
the  digits  are  moved  forwards  or  backwards."  He  adds:  "I 
therefore  prefer  to  teach  my  pupils  in  my  own  way,  in  spite 
of  the  ancient  custom.  Those  who  wish  to  know  the  shorter 
method  should  adopt  this  modern  plan." 

Sakabe  classifies  quadratic  equations  according  to  three 
types,  much  as  such  Eastern  writers  as  Al-Khowarazmi  and 
Omar  Khayyam  had  done  long  before,  and  as  was  the  custom 
until  relatively  modern  times  in  Europe.  His  types  were  as 
follows: 

—  ax2  +  bx  +  c  =  o, 

ax*  +  bx  —  c  =  o, 
ax2  —  bx  +  c  =  o, 

and  for  these  he  gives  rules  that  are  equivalent  to  the  formulas 


and 


He  takes,  as  will  be  seen,  only  the  positive  roots,  neglecting 
the  question  of  imaginaries,  a  type  never  considered  in  pure 
Japanese  mathematics.1 

1  Seki    knew    that    there    are    equations    with   no   roots,   the    musho  shiki 
(equations   without   roots),   but   of  the   nature   of  the   imaginary  he  seems  to 

14 


210 


XI.  The  Opening  of  the  Nineteenth  Century. 


Among  his  one  hundred  ninety-six  problems  is  one  in  Book  VI 
to  find  the  smallest  circle  that  can  be  touched  internally  by  a 
given  ellipse  at  the  end  of  its  minor  axis,  and  the  largest  one 
that  can  be  touched  externally  by  a  given  ellipse  at  the  end 
of  its  major  axis.  To  solve  the  latter  part  he  takes  a  sphere 
inscribed  in  a  cylinder  and  cuts  it  by  a  plane  through  a  point 
of  contact,  and  concludes  that  the  diameter  of  the  maximum 
circle  is  #2  — '—  b,  where  a  is  the  minor  axis  and  b  is  the 
major  axis.  For  the  other  case  he  finds  the  diameter  to  be 
&*  -—-  a. 

Sakabe  gives  some  attention  to  indeterminate  equations. 
Thus  in  solving  (Problem  104)  the  equation 

2#2  +  72  =  Z2 

he  takes  any  even  number  for  x  and  separates  —  x*  into  two 
factors,  m  and  n,  then  taking 

y  =  m  —  ;/,  z  =  m  +  n. 


Among  the  geometric  problems  is  the  following  (No.  138): 
"There  is  a  triangle  which  is  divided  into  smaller  triangles  by 
oblique  lines  so  drawn  from  the  vertex  that  the  small  inscribed 
circles  as  shown  in  the  figure  are  all  equal.  Given  the  altitude 
k  of  the  triangle  and  the  diameter  d  of  the  circle  inscribed 

have  been  ignorant.  In  Kawai's  Kaishlki  Shimpo  (1803)  the  statement  is 
made  that  there  may  be  a  mitsJw  (without  root),  that  is,  a  root  that  is 
neither  positive  nor  negative,  but  nothing  is  said  as  to  the  nature  of  such 
a  root. 


XI.  The  Opening  of  the  Nineteenth  Century.  211 

in  the  triangle,  required  to  find  the  diameter  of  one  of  the  n 
equal  circles."    His  solution  may  be  expressed  by  the  formula 


where  x  is  the  required  diameter. 

In  his  Book  X  Sakabe  gives  some  interesting  methods  of 
summing  a  series,  but  none  that  involved  any  principle  not 
already  known  in  Japan  and  in  the  world  at  large.  They  include 
the  general  plan  of  breaking  simple  series  into  partial  geometric 
series,  as  in  this  case: 


s  =  i  +  2r  +  3/-a  +  4^  + 

=  i  +     r  +  r2  +    r*  + 

+    r  +  r2  +    r*  + 

+  r2  +    7-3  + 


In  the  same  way  he  sums 
i  +     r  +   6r2  + 


557-4  +  . 
and  so  on,  these  including  the  general  types 


i  =  co    X-=  /'  .  /:=  oo     k  =  z' 

".    2  (2 


I  —  r 
r3 


/- 

(/ 


+ 


14* 


212  XI.  The  Opening  of  the  Nineteenth  Century. 

In  the   extraction   of  roots  Sakabe  gives    (Problem    167)  a 

« 

rule  for  the  evaluation  of  V N  that  has  some  interest.  He 
takes  any  number  a^  such  that  a"  is  approximately  equal 

to  N.    From  this  he  obtains  a2  =  N-—a"~I.    Then  the  real 
H 

value  of  VN  will  evidently  lie  between  at  and  a2,  so  that  he 
takes  for  his  third  approximation  a3  =  —  (a,.  +  a2),  increasing 

M  

or  decreasing  this  slightly  if  it  is  known  that  YN  lies  nearer 
#!  or  a2,  respectively.  He  next  calculates  #4  =  N -: -  a3"-1, 
and  continues  this  process  as  far  as  desired.  Thus,  to  find 

5  

1/0.125,  let  us  take  ax  =  0.66.    Then  we  find 
0a  =  0.6597541, 

*3=  0.6597539553865 

where  a2  is  correct  to  5  decimal  places  and  a3  to  12  decimal 
places. 

Sakabe    gives    many  other    interesting    problems,    including 

various  applications  of  the  yenri.  Among  his  results  is  the 
following  series: 

^L  =  I_jL_      *-4  (i. 3). (4- 6)         (i. 3. 5). (4. 6. 8) 

4  "  5        5-7-9       5-7-9-H.I3        5-7 15-17 

He  also  treats  of  the  length  of  the  arc  in  terms  of  the  chord 
and  altitude,  as  several  writers  had  already  done  in  the  pre- 
ceding century,  and  he  was  the  first  Japanese  to  publish  rules 
for  finding  the  circumference  or  an  arc  of  the  ellipse.1 

Sakabe  also  wrote  in  1803  a  work  entitled  the  Rippo  Eijiku,2 
in  which  he  treated  of  the  cubic  equation,  the  roots  being 
expressed  in  a  form  resembling  continued  fractions  which  in- 
volved only  square  roots.3  In  1812  he  published  his  Kwanki- 

1  Ajima  is  doubtfully  said  to  have  discovered  these  rules,  but  he  did  not 
print  them.     Sakabe   was   the   first   to  treat  of  the  ellipse  in  a  printed  work. 

2  Or  Rippo  Eichikic.    Literally,  Methods   of  approximating  by  increase  and 
decrease  (the  root  of)  a  cubic. 

3  This  work  was  never  printed.     The  same   plan  had  been  attempted  by 
one   Fujita   Seishin,   of  Tatebayashi    in  Joshu,   and  his   manuscript  had  been 


XI.  The  Opening  of  the  Nineteenth  Century.  2  1 3 

kodo-shohd?  a  work  on  spherical  trigonometry,  and  in  1816 
his  Kairo-anshinroku?  a  work  on  scientific  navigation. 

The  best-known  of  Sakabe's  pupils  was  Kawai  Kyutoku,3  a 
shogunate  samurai  of  high  rank  and  at  one  time  a  Superintendent 
of  Finance.  In  1803  Kawai  published  his  Kaishiki  ShimpoS 
although  it  is  suspected  that  Sakabe  may  have  had  a  hand  in 
writing  it.  He  records  in  the  preface  that  Sakabe  had  told  how 
in  his  day  some  European  and  Chinese  works  had  appeared 
in  Japan,  but  that  in  none  of  them  was  found  so  general  a 
method  as  he  himself  laid  before  his  pupils.  Indeed  there  was 
some  truth  in  this  boast,  since  the  subject  considered  was  the 
numerical  higher  equation,  and,  as  we  have  seen,  Horner's 
method  had  long  been  known  in  the  East.  It  was  here  that 
China  and  Japan  actually  led  the  world,  and  when  Sakabe 
and  Kawai  improved  upon  the  work  of  their  countrymen 
they  a  fortiori  improved  upon  the  rest  of  the  mathematical 
fraternity. 

This  improvement  consisted  first  in  abandoning  the  sangi  in 
favor  of  the  sorobanp  an  ideal  of  all  of  the  Japanese  mathe- 
maticians of  the  eighteenth  century.  In  the  second  place  the 
general  plan  of  work  was  simplified,  as  will  be  seen  from  the 
following  summary  of  the  process: 

Let  an  equation  of  the  #th  degree,  whose  coefficients  are 
integers,  either  positive  or  negative,  be  represented  by 

«!  +  a2  x  +  a^x2  +  ••  •  anxH~ "*  +  a>l+I  xn  =  o. 

The  n  roots  are  generally  positive  or  negative  according  as 
the  pairs  of  coefficients  (0W  +  1,  «„),  («„,  *„_,),  •  •  •  («a>  *,)  have 
different  signs  or  the  same  sign.  The  ^th  of  these  roots 
(r=  I,  2,  3  •  •  •  «)  may  be  found  as  follows: 

submitted  to  Sakabe,  who  found  it  so  complicated  that  he  proceeded  to 
simplify  it  in  this  work. 

1  Literally,  A  short  way  to  measure  spherical  arcs  by  the  telescopic  ob 
servation  of  heavenly  bodies. 

2  Literally,  The  safety  of  navigation. 

3  Or  Kawai  Hisanori. 

4  New  method  of  solving  equations. 

5  See  Kawai,  Kaishiki  Shimpo  (1803);   and   Endu,  Book  III,  p.  53. 


214  XI.  The  Opening  of  the  Nineteenth  Century. 

First  write 


Then  take 


and  let  B=~> 

A  may  be  assigned  any  value  so  long  as  P  shall  not  have 
a  different  sign  from  an_r,t  and  <2  sna'l  n°t  have  a  different 
sign  from  aM_r+a. 

Next  proceed  in  the  same  way  with  A',  denoting  the  result 
by  B'. 

If  now  we  shall  find  either  that 

A  >  B  and  A  <  B' 
or  that  A  <  B  and  A'  >  B'  , 

then  there  will  be  in  general  a  root  of  the  equation  between 
A  and  A'.  Now  by  narrowing  the  limits  between  which  the 
root  lies  a  first  approximation  may  be  reached,  but  it  suffices 
for  a  rough  approximation  to  take  the  average  of  A,  A',  B 
and  B'. 

Repeat  the  same  process  with  the  first  approximation  as 
was  followed  with  A  and  thus  obtain  a  second  approximation, 
and  so  on. 

For  example,  take  the  equation 

3360  —  2174*+  249*2  —  x*  =  o. 

Since  #3  and  «4  have  different  signs,  the  first  root  is  positive. 
Let  us  begin  with  A  =  10. 


XI.  The  Opening  of  the  Nineteenth  Century.  215 


Then  _  36, 

10 

336  -21/4  =-1838, 


., 

10 

-  183.8  +  249  =  65.2  =  p. 
Also  Q  =  —  i, 

so  that  B  =  —  ~  =  6$  .  2. 

Similarly  y2   =  10  £  =65.2 

/f  =  100  .5'  =  227 

^"  =  230  £"  =  239.6 

A'"=2$o  £"'=240.3, 

which  shows  that  the  first  root  lies  between  A"  and  A"',  since 

A"  <  B"  and  A"'  >  B'". 
Furthermore 

_  =  239.975,  or  nearly  240, 

which  is  the  first  approximation. 

In  the  same  way  the  approximate  second  root  is  7.21.  The 
rest  of  the  computation  is  along  lines  previously  known  and 
already  described. 

In  1820  an  architect  named  Hirauchi  Teishin1  published  a 
work  entitled  Sampo  Hengyd  Shinan?  and  later  the  Shoka 
Kiku  Ydkai,*  both  intended  for  men  of  his  profession  and  for 
engineers.  Much  use  is  made  of  graphic  computation,  as  in 
the  extraction  of  the  cube  root  by  the  use  of  line  intersections. 
In  1840  Hirauchi  wrote  another  work,  the  Sampo  Chokujutsu 
SeikaiS  in  which  he  treated  of  the  geometric  properties  of 

1  Also  known  by  his  earlier  name  of  Fukuda  Teishin. 

2  Also    transliterated    Sampo  -Henkei-  ski  nan.      Literally,    Treatise    on    the 
Hengyo  method,  Hengyo  meaning  the  changing  of  forms. 

3  Literally,  A  short  treatise  on  the  line  methods. 

4  Exact  notes  on  direct  mathematical  methods. 


2l6  XI.  The  Opening  of  the  Nineteenth  Century. 

figures  rather  than  of  their  mensuration.  While  the  book  had 
no  special  merit,  it  is  worthy  of  note  as  being  a  step  towards 
pure  geometry,  a  subject  that  had  been  generally  neglected 
in  Japan,  as  indeed  in  the  whole  East. 

It  often  happens  in  the  history  of  mathematics,  as  in  history 
in  general,  that  some  particular  branch  seems  to  show  itself 
spontaneously  and  to  become  epidemic.  It  was  so  with  algebra 
in  medieval  China,  with  trigonometry  among  the  Arabs,  with 
the  study  of  equations  in  the  sixteenth  century  Italian  algebra, 
and  with  the  calculus  in  the  seventeenth  century.  So  it  was 
with  the  study  of  geometry  in  Japan.  In  the  same  year  that 
Hirauchi  brought  out  his  first  little  work  (1820),  Yoshida  Juku 
published  his  Kikujutsu  Dzukai*  in  which  he  attempted  the 
solution  of  a  considerable  number  of  problems  by  the  use  of 
the  ruler  and  compasses.  It  is  true  that  this  study  had 
already  been  begun  by  Mizoguchi,  and  had  been  carried  on 
by  Murata  Koryu  under  whom  Yoshida  had  studied,  but  the 
latter  was  the  first  of  the  Mizoguchi  school2  to  bring  the 
material  together  into  satisfactory  form. 

About  this  time  there  lived  in  Osaka  a  teacher  named  Takeda 
Shingen,  who  published  in  1824  his  Sampo  Benran,*  in  which 
the  fan  problems  of  the  period  appear  (Fig.  42),  and  whose 
school  exercised  considerable  influence  in  the  western  provinces. 
He  also  wrote  the  Shingen  Sampo,  a  work  that  was  published 
by  his  son  in  1844.  The  old  epigram  which  he  adopted  "There 
is  no  reason  without  number,  nor  is  there  number  without 
reason,"  is  well  known  in  Japan. 

It  is,  however,  with  the  early  stages  of  geometry  that  we 
are  interested  at  this  period,  and  the  next  noteworthy  writer 
upon  the  subject  was  Hashimote  Shoho,  who  published  his 
Sampo  Tenzan  Shogakus/w*  in  1830.  The  particular  feature 


1  Illustrated  treatise    on    the    line    method.     His    works   are   thought    by 
some  to  have  been  written  by  Hasegawa. 

2  ENDO,  Book  III,  p.  91. 

3  Mathematical  methods  conveniently  revealed.     He   is  sometimes  known 
by  his  familiar  name,  Tokunoshin. 

4  Tenzan  method  for  beginners. 


XI.  The  Opening  of  the  Nineteenth  Century.  2 1/ 

of  interest  in  his  work  is  the  geometric  treatment  of  the  center 
of  gravity  of  a  figure.  One  of  his  problems  is  to  find  by 
geometric  drawing  the  center  of  gravity  of  a  quadrilateral, 
and  the  figure  is  given,  although  without  explanation.1 

This  problem  of  the  center  of  gravity  now  began  to  attract 
a  good  deal  of  attention  in  Japan.  Perhaps  the  first  real  study 2 
of  the  question  was  made  by  Takahashi  Shiji,  since  a  manu- 
script entitled  Toko  Scnsei  Chojutsu  Mokuroku*  mentions  a 
work  of  his  upon  this  subject.  Since  this  writer  was  acquainted 
with  the  Dutch  language  and  science,  he  doubtless  received 
his  inspiration  from  this  source.  His  son  Takahashi  Keiho4 
(1786-1830)  was,  like  himself,  on  the  Astronomical  Board  of 


Fig.  42.     From  Takeda  Shingen's  Sampo  Benran  (1824). 

the  Shogunate,  and  was  imprisoned  from  1828  until  his  death 
in  1830,  for  exchanging  maps  with  Siebold,  whose  work  is 
mentioned  in  Chapter  XIV. 

Of  the  other  minor  writers  of  the  opening  of  the  nineteenth 
century  the  most  prominent  was  Hasegawa  Kan,s  who  published 
his  Sampo  Shins  ho  (New  Treatise  on  Mathematics)  in  1830 

1  ENDO,    Book   III,    p.    107,    gives    a    conjectural   explanation.     He   is   of 
the   opinion   that  both   the   problem    and   the   solution   come    from    European 
sources. 

2  The  germ  of  the  theory  is  found  in  Seki's  writings. 

3  List  of  Master  Toko's  writings,  T6ko  being  his  nom  de  plume. 

4  Or  Takahashi   Kageyasu. 

5  Or  Hasegawa  Hiroshi. 


218 


XI.  The  Opening  of  the  Nineteenth  Century. 


under  the  name  of  one  of  his  pupils.  Hasegawa  Kan  was 
himself  a  pupil,  and  indeed  the  first  and  best-known  pupil,  of 
Kusaka  Sei,  the  same  who  had  studied  under  the  celebrated 
Ajima,  and  hence  he  had  good  mathematical  ancestry.  His 
work  was  a  compendium  of  mathematics,  containing  the 
soroban  arithmetic,  the  "Celestial  Element"  algebra,  the  tenzan 
algebra,  the  yenri,  and  a  little  work  on  geometry,  includ- 
ing some  study  of  roulettes  (Fig.  43).  So  well  written  was 
it  that  it  became  the  most  popular  mathematical  treatise  in 


Fig.  43.     From  Hasegawa  Kan's  Sampo  Shinsho  (1849  edition). 

the  country  and  brought  to  its  author  much  repute  as  a 
skilled  compiler.  Nevertheless  the  publication  of  this  work 
led  to  great  bitterness  on  the  part  of  the  Seki  school,  in- 
asmuch as  it  made  public  the  final  secrets  of  the  yenri  that 
had  been  so  jealously  preserved  by  the  members  of  this 
educational  sect.1  His  act  caused  his  banishment  from  among 
the  disciples  of  Seki,2  but  it  ended  the  ancient  regime  of  secrecy 


*  The  yenri  here  described  is  not  the  same  as  that  of  Ajima  or  Wada. 

*  ENDO  attributes   his    banishment  to  his   having  appropriated  to  his  own 
use  the  money  collected  for  printing  Ajima's  Fukyu  Sampo. 


XI.  The  Opening  of  the  Nineteenth  Century.  219 

in  matters  mathematical.  Hasegawa  died  in  1838  at  the  age 
of  fifty-six  years.1 

Among  the  noteworthy  features  of  the  Sampo  Shinsho 
mention  should  be  made  of  the  reversion  of  series*  in  one  of 
the  geometric  problems,  and  of  the  device  of  using  limiting 
forms  for  the  purpose  of  effecting  some  of  the  solutions.  One 
of  his  algebraic-geometric  problems  is  this:  Given  the  diameters 
of  the  three  escribed  circles  of  a  triangle  to  find  the  diameter 
of  the  inscribed  circle.  By  considering  the  case  in  which  the 
three  escribed  circles  are  equal,  as  one  of  the  limits  of  form, 
Hasegawa  gets  on  track  of  the  general  solution,  a  device  that 
is  commonly  employed  when  we  first  consider  a  special  case 
and  attempt  to  pass  from  that  to  the  general  case  in  geometry. 
The  principle  met  with  severe  criticism,  it  being  obvious  that 
we  cannot  reason  from  the  square  as  a  limit  back  to  a  rectangle 
on  the  one  hand  and  a  rhombus  on  the  other.  Nevertheless 
Hasegawa  was  very  skilful  in  its  use,  and  in  1835  he  wrote 
another  treatise  upon  the  subject,  the  Sampo  Kyoku-gyo  Shi- 
nan,^  published  under  the  name  of  his  pupil,'*  Akita  Yoshiichi 
of  Yedo. 

It  thus  appears  that  the  opening  years  of  the  nineteenth  century 
were  characterized  by  a  greater  infiltration  of  western  learning, 
by  some  improvement  in  the  tenzan  algebra,  and  by  the  initial 
steps  in  pure  geometry.  None  of  the  names  thus  far  mentioned 
is  especially  noteworthy,  and  if  these  were  all  we  should  feel 
that  Japanese  mathematics  had  taken  several  steps  backward. 
There  was,  however,  one  name  of  distinct  importance  in  the 
early  years  of  the  century,  and  this  we  have  reserved  for  a 
special  chapter, — the  name  of  Wada  Nei. 

1  Professor  Hayashi  gives  the   dates  1792-1832.     But   see  ENDO,  Book  II, 
p.  12,  and  KAWAKITA'S  article  in  the  Honcho  Siigakn  Koenshu,  p.  17. 

2  An  essentially  similar  problem,  in  connection  with  a  literal   equation  of 
infinite  degree,  seems  to  have  been  first  studied  by  Wada  Nei. 

3  Treatise  on  the  method  of  limiting  forms. 

4  A  custom  of  Hasegawa's.     See  the  note  on  Hirauchi,  above. 


CHAPTER  XII. 
Wada  Nei. 

It  will  be  recalled  that  in  the  second  half  of  the  eighteenth 
century  Ajima  added  worthily  to  the  yenri  theory,  bringing  for 
the  first  time  to  the  mathematical  world  of  Japan  a  knowledge 
of  a  kind  of  integral  calculus  for  the  quadrature  of  areas  and 
the  cubature  of  volumes.  The  important  work  thus  started  by 
him  was  destined  to  be  transmitted  through  his  pupil,  Kusaka 
Sei,1  to  a  worthy  successor  of  whom  we  shall  now  speak  at 
some  length. 

Wada  Yenzo  Nei  (i/S/'iS^),2  a  samurai  of  Mikazuki  in  the 
province  of  Harima,  was  born  in  Yedo.  His  original  name 
was  Koyama  Naoaki,  and  in  early  life  he  served  in  Yedo  in 
the  Buddhist  temple  called  by  the  name  Zojoji.  He  then 
changed  his  name  for  some  reason,  and  is  generally  known 
in  the  scientific  annals  of  his  country  as  Wada  Nei.  After 
leaving  the  temple  life  he  took  up  mathematics  under  the 
tutelage  of  Lord  Tsuchimikado,  hereditary  calendar-maker  to 
the  Court  of  the  Mikado.  He  first  studied  pure  mathematics 
under  a  certain  scholar  of  the  Miyagi  school,  and  then  under 
Kusaka  Sei.  As  has  already  been  mentioned,  this  Kusaka 
compiled  the  Ftikyu  Sampo  from  the  results  of  his  contact 
with  Ajima,  thus  bringing  into  clear  light  the  teaching  of  his 
master.  Although  it  must  be  confessed  that  he  did  not  have  the 
genius  of  Ajima,  nevertheless  Kusaka  was  a  remarkable  teacher, 

1  ENDO,  Book  III,  p.  127.     See  p.  172. 

2  KOIDE,  Yenri  Sankyo,  preface.     See  Chapter  XIV. 


XII.  Wada  Nei.  221 

giving  to  mathematics  a  charm  that  fascinated  his  pupils  and 
that  inspired  them  to  do  very  commendable  work.  Money 
had  no  attraction  for  him,  and  he  lived  a  life  of  poverty, 
dying  in  1839  at  the  age  of  seventy-five  years.1 

As  to  Wada,  no  book  of  his  was  ever  published,  and  all 
of  his  large  number  of  manuscripts,  which  were  in  the  keeping 
of  Lord  Tsuchimikado,  were  consumed  by  fire,2  that  great 
and  ever-present  scourge  of  Japan  that  has  destroyed  so  much 
of  her  science  and  her  letters.  Eking  out  a  living  by  fortune- 
telling  and  by  teaching  penmanship,  as  well  as  by  giving 
instruction  in  mathematics,3  selling  some  of  his  manuscripts  to 
gratify  his  thirst  for  liquor,  Wada's  life  had  little  of  happiness 
save  what  came  as  the  reward  of  his  teaching.  He  claimed 
to  have  had  among  his  pupils  some  of  the  most  distinguished 
mathematicians  of  his  day,4  men  who  came  to  him  to  learn 
in  secret,  recognizing  his  genius  as  an  investigator  and  as  a 
teacher.s 

It  will  be  recalled  that  Ajima  had  practiced  his  integration 
by  cutting  a  surface  into  what  were  practically  equal  elements 
and  summing  these  by  a  somewhat  laborious  process,  and 
then  passing  to  the  limit  for  n  =  oo.  In  a  similar  manner  he 
found  the  volumes  of  solids.  In  every  case  some  special  series 
had  to  be  summed,  and  it  was  here  that  the  operation  became 
tedious.  Wada  therefore  set  about  to  simplify  matters  by  con- 
structing a  set  of  tables  to  accomplish  the  work  of  the  modern 
table  of  integrals.  Since  his  expression  for  "to  integrate"  was 
the  Japanese  word  "to  fold"  (tatamu],  these  aids  to  calculation 
were  called  "folding  tables"  (jo-Jiyd),  and  of  these  he  is  known 


1  ENDO,  Book  III,   p.  121;   C.  KAWAKITA'S   article   in  the  Honcho  Sitgaku 
Koenshu,  p.  17;  KOIDE,  Yenri  Sankyo,  preface. 
*  KOIDE,  Yenri  Sankyo,  MS.  of  1842,  preface. 

3  ENDO,  Book  III,  p.  128. 

4  The    original   list  on   some    waste  paper  is   now   in   the   possession  of 
N.   Okamoto.      The   list   includes    the    names    of    Shiraishi,    Kawai,    Uchida, 
SaitO,  and  Ushijima,  with  many  others. 

5  See  also  ENDO,  Book  III,  p.  86. 


222 


XII.  Wada  Nei. 


to  have  left  twenty-one,  arranged  in  pamphlet  form  and  bearing 
distinctive  names.1 

In  1818  Wada  wrote  the  Yenri  Shinko  in  two  books,  published 
only  in  manuscript.  In  this  he  begins  by  computing  the  area 
of  a  circle  in  the  following  manner: 

The  diameter  is  first  divided  into  2n  equal  parts.  Then, 
drawing  the  lines  as  shown  in  the  figure,  it  is  evident  that 


a 


D,     D; 


d 

2n 


M 


D: 


M 


:ko: 


B 


and 
whence 


D  D '  = 

r       r  n    ' 


1  ENDO,  Book  III,  p.  74. 


XII.  Wada  Nei.  223 


Hence  twice  the  area  of  D  D"  N"    N 

r      r          i — i        > — i 


_d2  (     __  ^  __   I.H  1.37-6  _I_L3^S^8_  \ 

n     \  2n*        2-4«4        2.  4.  6«6        2.4.6.8^8 

Summing  for  r  =  I,  2,  3,  .  .  .  #,  we  have 

d*  f          i      *  i        "  \ 

77  v*  -  ^  2  r*  ~  ^^4  2  rl>       )' 


Multiplying,  and  then  proceeding  to  the  limit  for  n  =  oo,  we 
have  the  area  of  the  circle  expressed  by  the  formula 

a==d2(i—  —  -    _J_         *-3  1.3-5  \ 

2.3      2.4.5      2.4.6.7      2.4.6.8.9 

In  the  two  operations  of  summing  and  proceeding  to  the  limit 
Wada  makes  use  of  his  "folding  tables." 

By  a  similar  process  Wada  finds  the  circumference  to  be 


I*         I2. 


. 

3!         5!  7! 

and  he  obtains  formulas  for  the  area  of  a  segment  of  a  circle 
bounded  by  an  arc  and  a  chord,  or  by  two  arcs  and  two 
parallel  chords.1  It  is  also  said  that  he  gave  upwards  of  a 
hundred  infinite  series  expressing  directly  or  indirectly  the 
value  of  it,2  among  which  were  the  following: 


1  For  the  complete   treatment  see  HARZER,  P.,  loc.  cit.,  p.  33  of  the  Kiel 
reprint  of  1905.    HARZER  shows  that  the  formula  used  is  essentially  Newton's 
of  1666,  given  later  by  Wallis. 

2  ENDO,  A  short  account  of  the  progress  in  finding   the  value  of  n  in  Japan 
(in  Japanese),  in  the  Rigakkai,  vol.  Ill,  No.  4,  p.  24. 


224  XII.  Wada  Nei. 


945 


_  _ 

3.2       5.8  •    7-48       9-384       11-3840 

^  =    J_   •    JL  4.  _A_         *5     ,       I05      ,    __945_ 
4  "     3        5-2       7-8       9-48  T  11.384  ~  13.3840  "* 

«        JL   ,    _1_   ,  •  _3__   ,    _L5_          I05       ,        945 
8     "3         15-2       35-8    •    63.48  "r  99.384  T  143-3840 

«.  *.  i.  4.      L.  4.  -3-  4.      I5      4-       J°5—  4-  . 
32       15  T  35.2  f  63.8  ^  99.48  T  143-384  ^ 


±=  T  _   l   4.  A__I!  + I°i_J?45     ,    ... 
4  3         15       I05        945       10395  T 

TT       =  _i 3          ,    IS , i°5 ,    . 

2  1/2^   °  3-2-2  5-8-22     ">"    7.48.23  9. 384. 24  ^ 

the  larger  numbers  in  the  denominators  of  these  formulas  being 

2,  2.4,  2.4.6,   .  .. 

3.  3-5»  3-5-7,  ••  • 
i-3,  3-5,  5-7,  ••• 

The  same  principle  that  he  applies  to  the  circle  he  also  uses 
in  connection  with   the   ellipse,1   finding   the  perimeter  to  be2 


where  ;«  =  —  (  i ),   and   where  for  n=i   the  term  is  to 

4    N          a2  / 

be  taken  as  —  in. 

Wada  also  turned  his  attention  to  the  computation  of  volumes, 
simplifying  Ajima's  work  on  the  two  intersecting  cylinders,  and 
in  general  developing  a  very  good  working  type  of  the  integral 
calculus  so  far  as  it  has  to  do  with  the  question  of  men- 
suration. 

The  question  of  maxima  and  minima  had  already  been  con- 
sidered by  Seki  more  than  a  century  before  Wada's  time,  the 

1  In  his  Setsu-kei  Jun-gyakit. 

2  ENDO,  Book  III,  p.  81. 


XII.  Wada  Nei.  225 

rule  employed  being  not  unlike  the  present  one  of  equating,  a 
differential  coefficient  to  zero,  although  no  explanation  was 
given  for  the  method.  Naturally  it  had  attracted  the  attention 
of  many  mathematicians  of  the  Seki  school,  but  no  one  had 
ventured  upon  any  discussion  of  the  reasons  underlying  the 
rule.  The  question  is  still  an  open  one  as  to  where  Seki 
obtained  the  method.  In  the  surreptitious  intercourse  with  the 
West  it  would  be  just  such  a  rule  that  would  tend  to  find  its 
way  through  the  barred  gateway,  it  being  more  difficult  to 
communicate  a  whole  treatise.  At  any  rate  the  rule  was  known 
in  the  early  days  of  the  Seki  school,  and  it  remained  un- 
explained for  more  than  a  century,  and  until  Wada  took  up 
the  question.1  He  not  only  gave  the  .reason  for  the  rule,  but 
carried  the  discussion  still  further,  including  in  his  theory  the 
subject  of  the  maximum  and  minimum  values  of  infinite  series.2  ^— 
In  this  way  he  was  able  to  apply  the  theory  to  questions  in- 
volved in  the  yenri  where,  as  we  have  seen,  infinite  series  are 
always  found. 

In  1825  Wada  wrote  a  work  entitled  lyen  Sampo*  in  which 
he  treated  of  what  he  calls  "circles  of  different  species."  He 
says  that  "if  the  area  of  a  square  be  multiplied  by  the  moment 
of  circular  area4  it  is  altered5  into  a  circle,  and  we  have  the 
area  (of  this  circle).  If  the  area  of  a  rectangle  be  multiplied 
by  the  moment  of  circular  area  it  is  altered  into  an  ellipse, 
and  we  have  the  area  (of  this  ellipse).  If  the  volume  of  a 
cube  or  a  cuboid  be  multiplied  by  the  moment  of  the  spheri- 
cal volume,6  it  is  altered  into  a  sphere  or  a  spheroid,  and 
we  have  its  volume.  These  are  processes  that  are  well  known. 
It  is  possible  to  generalize  the  idea,  however,  applying  these 

1  It  is  found  in  his  manuscript  entitled  Tekijin  Ho-kyii-fw, 

2  ENDO,  Book  III,  p.  83. 

3  On  Circles  of  different  species. 

4  I.  e.,  by  — .     We  would  say,  a  =  Ttr*.     The  Japanese,  however,  always 
considered  the  diameter  instead  of  the  radius. 

5  This  seems  the  best  word  by  which  to  express  the  Japanese  form. 

6  I.  e.,  by  —  it. 


226 


XII.  Wada  Nei. 


processes  to  the  isosceles  trapezium,  to  the  rectangular  pyr- 
amid, and  so  on,  obtaining  circles  and  spheres  of  different 
forms." 

For  example,  given  an  ellipse  inscribed  in  the  rectangle 
ABCD  as  here  shown.  Take  YY'  the  midpoints  of  DC  and 
AB,  respectively  and  construct  the  isosceles  triangle  A  BY. 


Draw  any  line  parallel  to  AB  cutting  the  ellipse  in  P  and  Q, 
and  the  triangle  in  M  and  N,  as  shown.  Now  take  two 
points  Pe,  Q'  on  PQ,  symmetric  with  respect  to  YY',  and 
such  that  AB:MN=PQ:P'Q'.  Then  the  locus  of  P'  and 
Q'  becomes  a  curve  of  the  form  shown  in  the  figure,  touching 
AY  and  BY  at  their  mid-points  X'  and  X,  and  the  line  AB 


XII.  Wada  Nei.  227 

at  F'.  If  now  we  let  YY'  =  a,  and  X'X=b,  we  may  con- 
sider three  species  of  curve,1  namely  for  a~>b,  a  —  b,  a<ib. 

Wada  then  finds  the  area  inclosed  by  this  curve  to  be 
—  Tiafi,  the  process  being  similar  to  the  one  employed  for  the 
other  curvilinear  figures.  He  also  generalizes  the  proposition 
by  taking  an  isosceles  trapezium  instead  of  the  isosceles  triangle 
ABY,  the  area  being  found,  as  before,  to  be  —  nab,  where 
a  and  b  are  FF'  and  X' X  in  the  new  figure. 

Wada  also  devoted  his  attention  to  the  study  of  roulettes, 
being  the  first  mathematician  in  Japan  who  is  known  to  have 
considered  these  curves.  It  is  told  how  he  one  time  hung 
before  the  temple  of  Atago,  in  Yedo,  the  results  of  his  studies 
of  this  subject,  although  doing  so  in  the  name  of  one  of  his 
pupils.  The  problem  and  the  solution  are  of  sufficient  interest 
to  be  quoted  in  substantially  the  original  form.2 


"There  is  a  wheel  with  center  A  as  in  the  figure,  on  the 
circumference  of  which  is  the  center  of  a  second  wheel  B, 
while  on  the  circumference  of  B  is  the  center  of  a  third 

1  Wada  calls  these  the  seito-yen  (flourishing  flame-shaped  circle),  hoshu-yen, 
and  suito-yen  (fading  flame-shaped  circle). 

2  From  the  original.     See  also  ENDO,  Book  III,  p.  103. 

IS* 


228 


XII.  Wada  Nei. 


wheel,  C.  Beginning  when  the  center  C  is  farthest  from  the 
center  A,  the  center  B  moves  along  the  circumference  of  A, 
to  the  right,  while  the  center  C  moves  along  the  circum- 
ference of  B,  also  to  the  right,  the  motions  having  the  same 
angular  velocity  so  that  C  and  B  return  to  their  initial  positions 
at  the  same  time.  Let  the  locus  described  by  C  be  known 
as  the  ki-yen  (the  tortoise  circle).  Given  the  diameters  of  the 
wheels  A  and  B,  where  the  maximum  of  the  latter  should  be 
half  of  the  former,  required  to  find  the  area  of  the  ki-yen. 

"Answer  should  be  given  according  to  the  following  rule: 
Take  the  diameter  of  the  wheel  B;  square  it  and  double;  add 
the  square  of  the  diameter  of  A;  multiply  by  the  moment  of 
the  circular  area,  and  the  result  is  the  area  of  the  ki-yen. 

"A  pupil  of  Wada  Yenzo  Nei,  the  founder  of  new  theories 
in  the  yenri,  sixth  in  succession  of  instruction  in  the  School 
of  Seki."1 

Wada's  work  in  the  domain  of  maxima  and  minima  was 
carried  on  by  a  number  of  his  contemporaries  or  immediate 


B 


successors,  among  whom  none  did  more  for  the  theory  than 
Kemmochi  Yoshichi  Shoko.  His  contribution*  to  the  subject 
is  called  the  Yenri  Kyoku-su  Shokai  (Detailed  account  of  the 


1  The  rule  is  equivalent  to  saying  that  the  area  is  —  IT  (a*  -J-  2<$2),  where 
a  and  b  are  the  diameters  of  A  and  B.    Possibly  this  pupil  was  Koide  Shuki. 
Wada's  detailed  solution  is  lost. 

2  Unpublished,  and  exact  date  unknown. 


XII.  Wada  Nei.  229 

Circle-Principle  method  of  finding  Maxima  and  Minima),  and 
contains  two  problems.  The  first  of  these  problems  is  to  find 
the  shortest  circular  arc  of  which  the  altitude  above  its  chord 
is  unity.  For  this  he  gives  two  solutions,  each  too  long  to 
be  given  in  this  connection.  His  second  problem  is  to  con- 
struct a  right  triangle  ABC  with  hypotenuse  equal  to  unity, 
such  that  the  arc  A  A'  described  with  C'  as  a  center,  as  in 
the  figure,  shall  be  the  maximum,  and  to  find  the  length  of 
this  maximum  arc.1 

1  In  KEMMOCHI'S  work  there  are  certain  transcendental  equations  which 
are  solved  by  an  approximation  method  known  in  Japan  by  the  name  Kanrui- 
jutsii,  possibly  due  to  SaitS  Gigi  or  his  father.  Kemmochi  certainly  learned 
it  from  him.  He  also  wrote  a  work  usually  attributed  to  Iwai  Juyen,  the 
Sampo  yenri  hio  shaku,  one  of  the  first  to  explain  the  Kwatsu-jutsu  method. 
It  should  be  mentioned  that  the  cycloid  had  been  considered  before  Wada's 
time  by  Shizuki  Tadao,  who  discussed  it  in  his  Rekisho  Shinsho  (1800). 


CHAPTER  XIII. 
The  Close  of  the  Old  Wasan. 

Having  now  spoken  of  Wada's  notable  advance  in  the  yenri 
or  Circle  Principle,  in  which  he  developed  an  integral  calculus 
that  served  the  ordinary  purposes  of  mensuration,  there  remains 
a  period  of  activity  in  this  same  field  between  the  time  in 
which  he  flourished  and  the  opening  of  Japan  to  foreign  com- 
merce, which  period  marks  the  close  of  the  old  wasan,  or 
native  mathematics.  Part  of  this  period  includes  the  labors 
of  some  of  Wada's  contemporaries,  and  part  of  it  those  of  the 
next  succeeding  generation,  but  in  no  portion  of  it  is  there 
to  be  found  a  genius  such  as  Wada.  It  was  his  work,  his 
discoveries,  his  teaching  that  inspired  two  generations  of  mathe- 
maticians with  the  desire  to  further  improve  upon  the  Circle 
Principle.  We  have  seen  how  the  story  is  told  that  the  best 
mathematicians  of  his  day  went  to  him  in  secret  for  the 
purpose  of  receiving  instruction  or  suggestions,  and  it  is  further 
related  that  his  range  of  discoveries  was  greater  than  his  regular 
pupils  knew,  and  that  some  of  these  discoveries  appear  as  the 
work  of  others.  This  is  mere  rumor  so  far  as  any  trust- 
worthy evidence  goes  to  show,  but  it  lets  us  know  the  high 
estimate  that  was  placed  upon  his  abilities. 

Among  his  contemporaries  who  gave  serious  attention  to 
the  yenri  was  a  merchant  of  Yedo  by  the  name  of  lyezaki 
Zenshi  who  published  a  work  in  two  parts,  the  Gomel  Sampo, 
of  which  the  first  part  appeared  in  1814  and  the  second  in 
1826.  There  is  a  charming  little  touch  of  Japan  in  the  fact 
that  many  of  the  problems  relate  to  figures,  and  in  particular 
to  groups  of  ellipses,  that  can  be  drawn  upon  a  folding  fan, 
that  is,  upon  a  sector  of  an  annulus. 


XIII.  The  Close  of  the  Old  Wasan.  23  I 

lyezaki  gives  also  some  problems  in  the  yenri  of  a  rather 
advanced  nature.  For  example,  he  gives  the  area  of  the 
maximum  circular  segment  that  can  be  inscribed  in  an  isosceles 
triangle  of  base  b  and  so  as  to  touch  the  equal  sides  s,  as 


He  also  states  that  if  an  arc  be  described  within  a  right 
triangle,  upon  the  hypotenuse  as  the  chord,  and  if  a  circle  be 
drawn  touching  this  arc  and  the  two  sides  of  the  triangle,  the 
maximum  diameter  of  this  circle  is 


where  a,  b  and  c  are  the  sides. 

Contemporary  with  lyezaki,    or   immediately   following  him, 
were  several  other  writers  who  paid  attention  to  figures  drawn 


Fig.  44.     From  Yamada  Jisuke's  Sampo  Tenzan  Shinan 
(Bunkwa  era,   1804 — 1818). 

upon  fans.  Among  these  may  be  mentioned  Yamada  Jisuke 
whose  Sampd  Tenzan  Shinan  (Instructor  in  the  tenzan  mathe- 
matics) appeared  early  in  the  century  (see  Fig.  44);  Takeda 
Tokunoshin  whose  Kaitei  Sampd  appeared  in  1818  (see  Fig.  45); 
Ishiguro  Shin-yu  (see  Fig.  46),  already  mentioned  in  Chapter  V 


232  XIII.  The  Close  of  the  Old  Wasan. 

as  the  last  Japanese  writer  to  make  much  of  the  practice  of 
proposing    problems    for    his    rivals    to    solve;    and    Matsuoka 


Fig.  45.     From  Takeda  Tokunoshin's  Kaitei  Sampo  (1818). 


Fig.  46.     Tangent  problem  from  Ishiguro  Shin-yu  (1813). 


XIII.  The  Close  of  the  Old  Wasan. 


233 


Yoshikazu,  whose  Sangaku  Keiko  Daizen,  an  excellent  com- 
pendium of  mathematics,  appeared  in  1808  and  again  in  1849. 

Also  contemporary  with  lyezaki  was  Shiraishi  Chochu  (1796- 
1862)  who  published  a  work  entitled  SJiamei  Sampu*  in  1826. 
He  was  a  samurai  in  the  service  of  Lord  Shimizu,  a  near 
relative  of  the  Shogun.  While  most  of  the  problems  in  this 
treatise  relate  to  the  yenri,  there  is  some  interesting  work  in 
the  line  of  indeterminate  equations.  One  of  these  equations 
bears  the  name  of  Gokai  Ampon,  and  like  the  rest  was  hung 
before  some  temple.  The  problem  is  as  follows: 

"There  are  three  integral  numbers,  heaven,  earth,  and  man, 
which  being  cubed  and  added  together  give  a  result  of  which 
the  cube  root  has  no  decimal  part.  Required  to  find  the 
numbers." 

The  problem  is,  of  course,  to  solve  the  equation  x*  + j3  +  z* 
=  #3  in  integers.  The  solution  is  given  in  Gokai's  name,  and 
he  is  known  to  have  been  an  able  mathematician,  but  whether 
it  was  his  or  Shiraishi's  is  unknown.  In  a  manuscript  com- 
mentary on  the  work2  the  following  discussion  of  the  equation 
appears: 

First  a  table  is  constructed  as  follows: 


I3  + 

7  = 

23 

123  + 

469=  133 

23  + 

19  = 

33 

133  + 

547=  143 

33  + 

37  = 

43 

I43  + 

631  =  153 

43  + 

61  = 

53 

i53  + 

721  =  163 

53  + 

91  = 

63 

l63  + 

817=  173 

i73  + 

r\j  •}       -     r83 

913  —     10° 

o3  + 

T  1    _l_ 

127  = 

73 

S3 

I  83  + 

1027  =     193 

7j  + 

3 

83  + 
03  + 

217  = 
271  — 

93 
IQ3 

533  + 

8587  =     543 

IQ3  + 

113 



113  + 

397  = 

123 

1023+31519=1033 

1  Mathematical  Results  hung  in  Temples. 

2  Shamei  Sampu  Kaigi. 

3  In  the  table  these  missing  numbers  are  given,  but  they  are  not  necessary 
for  our  purposes. 


234  xni-  The  CIose  of  the  Old  VVasan. 

Taking  the  second  terms,  7,  19,  37,  .  .  .,  it  will  be  seen  that 
the  successive  differences  are  as  follows: 

7  19  37  61  91  127 

12  1  8  24  30  36 

6666 

We  can  thus  easily  pick  out  the  numbers  that  are  the  sums 
of  two  cubes,  such  as  91  =  33  +  43,  1027  =  33  4-  io3,  and  so 
on,  and  frame  the  corresponding  relations  as  has  been  done 
in  the  table,  adding  others  at  will,  such  as 

I973  +  117019=  I983 
3O63  +  281827  =  307  3. 
Then  writing  n=y+  i, 

from  A'3  +  jF3  +  z*  =  «3 

we  can  derive 


Then  writing  the  selected  equalities  in  the  form 

43  +    53+33=    63  3i3  +  iO23  H-  i23= 

IQ3  +  1  83  +   33  =  193  463  +  1973  +  273  = 

193  +  533  +  123  =  543  643  +  306^  +  273  ==  3075 

we  notice  that  our  values  of  x,  y,  z,  and  n  may  be  expressed 
as  follows: 

*3.i  +  i  3-3  +  1  3-6+1  3.10+1  3.15  +  1  3-2i  +  i 

y    5  18  53  102  197  306 

a    3-12  3-i2  3-22  3-22  3-32  3-32 

n   6  19  54  103  198  307 

We  therefore  see  that  z  is  of  the  form  3«2.     Corresponding 
to  this  value  of  z,  x  is  of  the  form 


where  r=  2«—  I   or  2  a,  alternately.     That  is, 
x  =  6a2  +  3«  +  i. 


XIII.  The  Close  of  the  Old  Wasan. 


235 


Substituting  these  values  in  (i)  we  have 

324<26  +  432^5  +  360^+  i8o#3  +  6oa*±_  \2a  +  i 

=  4j2  +  47  +  i» 
from  which 

y  =  9^3  +  6a2  +  $a,  or  ga*  —  6a2  +  $a  —  i, 
and    n  =  y  +  i  =  9#3  +  6<22  +  30+1,  or  9^3_6rt2  +  30, 

which  gives  the  general  solution. 

Among  the  geometric  problems  given  by  Shiraishi  two,  given 
in  Ikada's  name,  may  be  mentioned  as  types. 


The  first  is  as  follows:  "An  ellipse  is  inscribed  in  a  rectangle, 
and  four  circles  which  are  equal  in  pairs  are  described  as 
shown  in  the  figure,  A  and  B  touching  the  ellipse  at  the  same 
point.  Given  the  diameters  (a  and  b}  of  the  circles,  required 
to  find  the  minor  axis  of  the  ellipse."  The  result  is  given  as 
a  +  b  +  V(2a  +  d)  b. 

The  second  problem  is  to  find  the  volume  cut  from  a  sphere 
by  a  regular  polygonal  prism  whose  axis  passes  through  the 
center  of  the  sphere. 

There  are  also  two  problems  given  as  solved  by  Shiraishi's 
pupils  Yokoyama  and  Baishu,  of  which  one  is  to  find  the  volume 


236 


XIII.  The  Close  of  the  Old  Wasan. 


cut  from  a  cylinder  by  another  cylinder  that  intersects  it 
orthogonally  and  touches  a  point  on  the  surface,  and  the 
other  is  to  find  the  volume  cut  from  a  sphere  by  an  elliptic 
cylinder  whose  axis  passes  through  the  center. 

The  Shamei  Sampu  contains  a  number  of  problems  of  this 
general  nature,  including  the  finding  of  the  spherical  surface 
that  remains  when  a  sphere  is  pierced  by  two  equal  circular 
cylinders  that  are  tangent  to  each  other  in  a  line  through  the 


Fig.  47.     From  Iwai  Juyen's  Sampo  Zasso  (1830). 

center  of  the  sphere;  the  finding  of  the  area  cut  from  a 
spherical  surface  by  a  cylinder  whose  surface  is  tangent  to 
the  spherical  surface  at  one  point;  the  finding  of  the  volume 
cut  from  a  cone  pierced  orthogonally  to  its  axis  by  a  cylinder, 
and  the  finding  the  surface  of  an  ellipsoid. 

Shiraishi   also  wrote   a  work    entitled  Suri  Mujinzo*  but  it 

1  An  inexhaustible  Fountain    of   Mathematical  Knowledge.     It  is  given  in 
Ikeda's  name. 


XIII.  The  Close  of  the  Old  Wasan. 


237 


was  never  printed.  It  is  a  large  collection  of  formulas  and 
relations  of  a  geometric  nature.  His  pupil  Kimura  Shoju 
published  in  1828  the  Onchi  Sanso  which  also  contained 


T    ^  -• 


Fig.  48.     From  Aida  Yasuaki's  Sampo  Ko-kon  Tsiiran. 


238  XIII.  The  Close  of  the  Old  Wasan. 

numerous  problems  relating  to  areas  and  volumes.  Interesting 
tangent  problems  analogous  to  those  given  by  Shiraishi  are 
found  in  numerous  manuscripts  of  the  nineteenth  century. 
Illustrations  are  seen  in  Figs.  50  and  51,  from  an  undated 
manuscript  by  one  Ivvasaki  Toshihisa,  and  in  Fig.  48,  from 
a  work  by  Aida  Yasuaki. 

Another  work  applying  the  yenri  to  mensuration,  the  Sampo 
Zasso,  by  Iwai  Juyen  (or  Shigeto),  appeared  in  1830.  Iwai 
was  a  wealthy  farmer  living  in  the  province  of'Joshu  and  he 
had  studied  under  Shiraishi.  He  also  gives  the  problem  of 
the  intersecting  cylinders  (see  Fig.  47),  and  the  problem  of 
finding  the  area  of  a  plane  section  of  an  anchor  ring.  In 


Fig.  49.     From  Hori-ike's  Yomw  Sampo  (1829). 

1837  Iwai  published  a  second  work  entitled  Yenri  Hyoshaku? 
although  it  is  said  that  this  was  written  by  Kemmochi  Yoshichi. 
In  this  the  higher  order  of  operations  of  the  yeuri  were  first 
made  public,  and  some  notion  of  projection  appears.  Another 
work  published  in  the  same  year,  the  Keppi  Sampo  by  Hori- 
ike  Hisamichi,  resembles  it  in  these  respects.  Hori-ike's  Yo- 
mio  Sampo  (1829)  contains  some  interesting  fan  problems 
(see  Fig.  49). 

More  talented  as  a  mathematician,  however,  and  much  more 
popular,  was  Uchida  Gokan,2  who  at  the  age  of  twenty-seven 


1  The  Method  of  the  Circle  Principle  explained. 
1  Or  Uchida  Itsumi. 


XIII.  The  Close  of  the  Old  Wasan. 


239 


Fig.  50.     Tangent  problem,  from  a  manuscript  by  Iwasaki  Toshihisa. 


240 


XIII.  The  Close  of  the  Old  Wasan. 


published  a  work  that  brought  him  at  once  into  prominence. 
Uchida  was  born  in  1805  and  studied  mathematics  under 
Kusaka,  taking  immediate  rank  as  one  of  his  foremost  pupils. 
In  1832  he  published  his  Kokon  Sankan'1  in  two  books  which 
included  a  number  of  problems  that  were  entirely  new,  and 
did  much  to  make  the  higher  yenri.  Sections  of  an  elliptic 
wedge,  for  example,  were  new  features  in  the  mathematics  of 
Japan,  and  the  following  problems  showed  his  interest  in  the 
older  questions  as  well: 


There  is  a  rectangle  in  which  are  inscribed  an  ellipse  and 
four  circles  as  shown  in  the  figure.  Given  the  diameters  of 
the  three  circles  A,  B  and  C,  viz.,  a,  b  and  c,  it  is  required 
to  find  the  diameter  of  the  circle  D. 

The  rule  given  is  as  follows:  Divide  a  and  b  by  c,  and  take 
the  difference  between  the  square  roots  of  these  quantities. 
To  this  difference  add  i  and  square  the  result  This  multiplied 
by  c  gives  the  diameter  of  D.  This  rule  was  suspected  by 
the  contemporaries  and  the  immediate  successors  of  Uchida, 
but  they  were  unable  to  show  that  it  was  false.*  Uchida  was, 

i  Mirror  (model)  of  ancient  and  modern  Mathematical  Problems. 
*  For  this  information  the  authors  are  indebted  to  T.  HAGIWARA,  the  only 
survivor,  up  to  his  death  in  1909,  of  the  leaders  of  the  old  Japanese  school. 


XIII.  The  Close  of  the  Old  Wasan. 


241 


however,  aware  of  it,  although  it  appears  in  none  of  his 
writings.1  Uchida  also  gave  several  interesting  fan  problems 
(see  Fig.  55). 

Uchida  died  in  1882,  having  contributed  not  unworthily 
to  mathematics  by  his  own  writings,  and  also  through  the 
works  of  his  pupils.1  Among  the  latter  works  are  Shino 
Chikyo's  Kakki  Sampo  (1837),  Kemmochi's  Tan-i  Sampo  (1840) 


t 


Fig.  51.     Problem    of  spheres   tangent   to    a  tetrahedron,   from   a  manuscript 
by  Iwasaki  Toshihisa. 

and  Sampo  Kaiwun  (1848),  Fujioka's  Sampo  Yenri-tsu  (1845), 
Takenouchi's  Sampo  Yenri  Kappatsu  (1849)  and  Kuwamoto 
Masaaki's  Sen-yen  Kattsu  (1855),  not  to  speak  of  several  others. 

1  This    information    is    communicated    to    us    by    C.  KAWAKITA,    one    of 
Uchida's  pupils. 

2  C.  KAWAKITA'S    article    in    the    Honcho  Sugaku  Koen-shii,    1908,    p.  20. 
Shino  Chikyo's  nom  de  plume  was  Kenzan.     • 

16 


242 


XIII.   The  Close  of  the  Old  Wasan. 


Among  the  contemporaries  of  Wada  should  also  be  men- 
tioned Saito  Gigi,  whose  Yenri-kan  appeared  in  1834.  It  is 
possible  that  the  real  author  was  Saito's  father,  Saito  Gicho 
(1784-1844),  who  also  took  much  interest  in  mathematics. 
Father  and  son  were  both  well-to-do  farmers  in  Joshu  with 
whom  mathematical  work  was  more  or  less  of  a  pastime.  The 
Yenri-kan  deserves  this  passing  mention  on  account  of  the 
fact  that":  it  contains  a  problem  on  the  center  of  gravity,  and 
several  problems  on  roulettes. 


Fig.  52.     From  Kobayashi's  Sampo  Koren  (1836). 

In  1836  appeared  Kobayashi  Tadayoshi's  Sampo  Koren  in 
which  is  considered  the  volumes  of  intersecting  cylinders  and  a 
problem  on  a  skew  surface.  The  latter  is  stated  as  follows: 
"There  is  a  'rhombic  rectangle'1  which  looks  like  a  rectangle 
when  seen  from  above,  and  like  a  rhombus  when  seen  from 
the  right  or  left,  front  or  back.  Given  the  three  axes,  required 
the  area  of  the  surface."  Here  the  bases  are  gauche  quadri- 
laterals. (The  drawing  is  shown  in  Fig.  52.)  Saito  also  published 
a  similar  work,  the  Yenri  Shinshin,  in  1840. 


1  This  is  the  literal  translation  of  choku  bishi.     The   figure  is    a  solid  and 
is  denned  in  the  problem. 


XIII.  The  Close  of  the  Old  Wasan. 


243 


At  about  the  same  period  there  appeared  numerous  works 
of  somewhat  the  same  nature,  of  which  the  following  may  be 
mentioned  as  among  the  best: 

Gokai  Ampon's  (1796—1862)  Sampo  Semmon  Sho  (1840),  a 
work  on  the  advanced  tenzan  theory,  with  some  treatment  of 
magic  squares  (Fig.  54). 


*P  S  #  f 

V        A        «?         V 


-^fc 


v\,.      JS 


0 


*t 


•^  : 


Fig-  S3-     From  Murata's  Sampo  Jikata  Shinan  (1835). 

Yamamoto  Kazen's  Sampo  Jojutsu'1  (1841),  containing  an 
extensive  list  of  formulas  and  excellent  illustrations  of  the 
problems  of  the  day  (see  Fig.  57). 

Murata  Tsunemitsu's  Sokuyen  Shokai  (1833),  relating  to  the 
tenzan  algebra  applied  to  the  ellipse,  and  his  Sampo  Jikata 
Shinan  (1835),  dealing  with  enginering  problems  (Fig.  53). 
Murata's  pupil  Toyota  wrote  the  Sampo  Dayen-kai  in  1842, 
also  relating  to  the  tenzan  algebra  applied  to  the  ellipse.2 

1  Aids  in  Mathematical  Calculation. 

2  Besides  Murata's  work  we  have  consulted  ENDO,  Book  III,  p.  129. 

16* 


244 


XIII.  The  Close  of  the  Old  Wasan. 


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Fig.  54.     Magic  Squares  from  Gokai's  Sampo  Semmon  Sho  (1840). 

A  work  by  a  Buddhist  priest,  Kakudo  written  in  Kyoto 
in  1794  and  published  in  1836,  entitled  Yenri  Kiku  Sampo, 
giving  a  summary  of  the  yenri. 

Chiba  Tanehide's  Sampo  Shin-sho  (1830),  a  large  compendium 
of  mathematics,  actually  the  work  of  Hasegawa  Kan. 


XIII.  The  Close  of  the  Old  Wasan. 


245 


Fig.  55.     From  Uchida's  Kokon  Sankan  (1832). 

The  Sampo  Tenzan  Tebikigusa,  of  which  the  first  part  was 
published  by  Yamamoto  in  1833  and  the  second  part  by 
Omura  Isshu  (1824 — 1891)  in  1841.  This  was  a  treatise  on 


Fig.  56.     From  Minami's  Sampo  Yenri  Sandai  (1846). 


246 


XIII.  The  Close  of  the  Old  Wasan. 


tenzan  algebra.    Some  of  the  fan  problems  in  this  work  are  of 
considerable  interest.     (See  Fig.  58.) 

Kikuchi  Choryo's   Sampo   Seisu  Kigensho  (1845),  a  treatise 
on  indeterminate  analysis. 


t 


Fig.  57.     From  Yamamoto  Kazan's  Sampo  Jojutsu  (1841). 


XIII.  The  Close  of  the  Old  Wasan. 


247 


Minami  Ryoho's  Sampo  Yenri  Sandai  (1846),  with  some 
treatment  of  roulettes  (see  Fig.  56)  and  the  Juntendo  Sampu* 
(1847)  by  Iwata  Seiyo  and  Kobayashi  (not  Tadayoshi).  Curi- 
ously, -the  first  ten  pages  of  Minami's  work  are  numbered  with 
Arabic  numerals. 

Kaetsu's  Sampo  Yenri  Katsund  (1851),  a  work  on  the  higher 
yenri.  This  was  considered  of  such  merit  that  it  was  reprinted 
in  China. 

Iwasaki  Toshihisa's    Yachu  sak  kai  (1831),  Saku  yen  riu  kwai 


7 
f 

v^ 

^/ 

. 

Jfr. 

* 

-tr 

1        : 

*                         ty         $          ^ 
£         &         ^         ®        & 

Fig.  58.     From  Yamamoto  and  Omura  Isshu's  SampO  Tenzan  Tebikigusa 

(1833,   184I). 

Juntendo  Mathematical  Problems. 


248  XIII.  The  Close  of  the  Old  Wasan. 

gi,  and  Shimpeki  sampo,  all  works  of  considerable  merit  in  the 
line  of  geometric  problems. 

Baba  Seito's  Shi-satsu  Henkai  (1830),  generally  known  by 
the  later  title  Sampo  Kisho. 

Hasegawa  Ko's  Kyuseki  Tsuko*  (1844),  published  under  the 
name  of  his  pupil  Uchida  Kyumei.  This  is  more  important 
than  the  works  just  mentioned.  It  consists  of  five  books  and 
gives  a  very  systematic  treatment  of  the  yenri,  beginning 
with  the  theory  of  limits  and  the  use  of  the  "folding  tables" 
of  Wada  Nei.  It  treats  of  the  circular  wedge  and  its  sections, 
of  the  intersections  of  cylinders  and  spheres  (see  Fig.  59),  of 
ovals,  or  circles  of  various  classes,  as  studied  by  Wada,  and 
also  of  the  cycloid  and  epicycloid. 

The  study  of  the  catenary  begins  about  1860.  The  first  to 
give  it  attention  were  Omura  and  Kagami,  but  the  first  printed 
work  in  which  it  is  discussed  is  the  Sampo  Hoyen-kan  (1862) 
of  Hagiwara  Teisuke  (1828 — 1909).  Another  interesting  problem 
which  appears  in  this  work  is  that  of  the  locus  of  the  point  of 
contact  of  a  sphere '  and  plane,  the  sphere  rolling  around  on 
the  plane  and  always  touching  an  anchor  ring  that  is  normal 
to  and  tangent  to  the  plane.  Hagiwara  also  published  a  work 
entitled  Sampo  Yenri  Shiron  (1866)  in  which  he  corrected  the 
results  of  thirty-four  problems  given  in  twenty-two  works 
published  at  various  dates  from  the  appearance  of  Arima's 
Shuki  Sampo  (1769)  to  his  own  time  (see  Figs.  60,  6 1).  He  also 
published  a  work  entitled  Yenri  San-yo  (1878),  the  result  of 
his  studies  of  the  higher  yenri  problems.  His  manuscript  called 
the  Reikan  Sampo  was  published  in  1910  through  the  efforts 
of  a  number  of  Japanese  scholars.  /  Hagiwara  was  born  in 
1828,  and  was  a  farmer  in  narrow  circumstances  in  the 
province  of  Joshu.  Not  until  about  1854  did  he  take  an 
interest  in  mathematics,  but  when  he  recognized  his  taste  for  the 
subject  he  became  a  pupil  of  Saito's,  traveling  on  foot  ten 
miles  on  the  eve  of  a  holiday  so  as  to  have  a  full  day  with 
his  teacher.  His  manuscripts  were  horded  in  a  miserly  fashion 

1  General  Treatment  of  Quadrature  and  Cubature. 


XIII.  The  Close  of  the  Old  Wasan. 


249 


M^i 


5*  it  4  si  >DM  >u 


W  -t- 

^    ^F 

^    ^   ^t    ^ 

0       -*F»      ^" 

J  ^°  n  ii 

-f-        ^ 
^        ^r 

-i 

^r 

*»;?> 
iPH] 

-» 

A« 

* 

^i, 

* 

•T 
« 
« 


^ 

^ 
«t 

^L 


^ 

4*r 
* 
^'j 


&$ 

^ 

« 
^ 


Fig-  59-     From  Hasegawa  Ko's  Kyuseki  Tsuko  (1844). 


250 


XIII.  The  Close  of  the  Old  Wasan. 


until  his  death,  November  28,  1909,  when  the  last  great  mathe- 
matician of  the  old  school  passed  away. 

Mention  should  be  made  at  this  time  of  the  leading  mathe- 
maticians who  were  the  contemporaries  of  Hagiwara,  and  who 
were  living  when  the  Shogunate  gave  place  to  the  Empire  in 
1868.  Of  these,  Hodoji  Wajuro  was  born  in  1820  and  died 
in  1 87 1.1  He  was  the  son  of  a  smith  in  Hiroshima,  and  although 
he  led  a  kind  of  vagabond  existence  he  had  a  good  deal  of 
mathematical  ability.  It  is  said  that  he  was  the  real  author 
of  Kaetsu's  Yenri  Katsuno.  Several  other  books  are  known 
to  have  been  written  by  him,  but  they  were  not  published 
under  his  own  name. 

Iwata  Kosan  (1812 — 1878),  born  a  samurai,  devoted  his 
attention  particularly  to  the  ellipse.  The  following  is  his  best 

known  problem: 

Given    an    ellipse   E  tangent  to 

two  straight  lines  and  to  four 
circles,  A,  B,  C,  D,  as  shown  in 
the  figure.  Given  the  diameters 
of  A,  B  and  C,  required  to  find 
the  diameter  of  D.  His  solution, 
given  in  1866,  is  essentially  the  pro- 
portion a\b  =  c:d,  where  a,  b,  c, 
d  are  the  respective  diameters  of 
A,  B,  C  and  D.  The  problem 
was  afterwards  extended  to  any 
four  conies  instead  of  four  circles, 
by  H.  Terao  and  others. 

Kuwamoto  Masaaki  wrote  the 
Senyen  Kattsu  in  1855,  and  in  it 
he  treated  of  roulettes  of  various 

kinds  (see  Fig.  62),  of  elliptic  wedges  (see  Fig.  63),  and  other 

forms  at  that  time  attracting  attention. 

Takaku  Kenjiro  (1821  — 1883)  wrote  the  Kyokusu  'I aisei-jutsu 

in  which  he  made  some  contribution  to  the  theory  of  maxima 

and  minima. 

1  C.  KAWAKITA,  in  the  Honcho  Sugaku  Koenshu,  p.  23. 


D 


XIII.  The  Close  of  the  Old  Wasan. 


251 


Fig.  60.     From  Hagiwara's  Sampo  Yenri  Shiran  (1866). 

Fukuda  Riken  (1815 — 1889)  lived  first  in  Osaka  and  finally 
in  Tokyo.  He  was  a  teacher  of  some  prominence,  and  his 
Sampo  Tamatebako  appeared  in  1879. 


H. 

jvvs 


Fig.  6l.     From  Hagiwara's  Sampo  Yenri  Shiran  (1866). 


252 


XIII.  The  Close  of  the  Old  Wasan. 


Yanagi  Yuyetsu  (1832 — 1891)  was  a  naval  officer  who  gave 
some  attention  to  the  native  Japanese  mathematics. 


Fig.  62.     From  Kuwamoto  Masaaki's  Sen  yen  Kattsfi  (1855). 

Suzuki  Yen,  who  may  still  be  living  wrote  a  work  (1878)  upon 
circles  inscribed  in  or  circumscribed  about  figures  of  various 
shapes. 


Fig.  63.     From  Kuwamoto  Masaaki's  Sen  yen  Kattsil  (1855). 


XIII.  The  Close  of  the  Old  Wasan.  253 

Thus  closes  the  old  wasan,  the  native  mathematics  of  Japan. 
It  seems  as  if  a  subconscious  feeling  of  the  hopelessness  of 
the  contest  with  Western  science  must  have  influenced  the 
last  half  century  preceding  the  opening  of  Japan.  There  was 
really  no  worthy  successor  of  Wada  Nei  in  all  this  period, 
and  the  feeling  that  was  permeating  the  political  life  of  Japan, 
that  the  day  of  isolation  was  passing,  seems  also  to  have 
permeated  scientific  circles.  With  the  scholars  of  the  country 
obsessed  with  this  feeling  of  hopelessness  as  to  the  native 
mathematics,  the  time  was  ripe  for  the  influx  of  Western 
science,  and  to  this  influence  from  abroad  we  shall  now  devote 
our  closing  chapter. 


CHAPTER  XIV. 
The  Introduction  of  Occidental  Mathematics. 

We  have  already  spoken  at  some  length  in  Chapter  IX  of 
the  possible  connection,  slight  at  the  most,  between  the  mathe- 
matics of  Japan  and  Europe  in  the  seventeenth  century.  The 
possibility  of  such  a  connection  increased  as  time  went  on, 
and  in  the  nineteenth  century  the  mathematics  of  the  West 
finally  usurped  the  place  of  the  wasan.  During  this  period 
of  about  two  centuries,  from  1650  to  the  opening  of  Japan  to 
the  world,  knowledge  of  the  European  mathematics  was  slowly 
finding  its  way  across  the  barriers,  not  alone  through  the 
agency  of  the  Dutch  traders  at  Nagasaki,  but  also  by  means 
of  the  later  Chinese  works  which  were  written  under  the  in- 
fluence of  the  Jesuit  missionaries.  These  missionaries  were 
men  of  great  learning,  and  they  began  their  career  by  im- 
pressing this  learning  upon  the  Chinese  people  of  high  rank. 
Matteo  Ricci  (1552 — 1610),  for  example,  with  the  help  of  one 
Hsu  Kiiang-chty£  (1562 — 1634),  translated  Euclid  into  the 
Chinese  language  in  1607,  and  he  and  his  colleagues  made 
known  the  Western  astronomy  to  the  savants  of  Peking.  It 
must  be  admitted,  however,  that  only  small  bits  of  this  learning 
could  have  found  a  way  into  Japan.  Euclid,  for  example,  seems 
to  have  been  unknown  there  until  about  the  beginning  of  the 
eighteenth  century,  and  not  to  have  been  well  known  for  two 
and  a  half  centuries  after  it  appeared  in  Peking. 

Some  mention  should,  however,  be  made  of  the  work  done 
for  a  brief  period  by  the  Jesuits  in  Japan  itself,  a  possible  in- 
fluence on  mathematics  that  has  not  received  its  due  share  of 


XIV.  The  Introduction  of  Occidental  Mathematics.  255 

attention.1  It  is  well  known  that  the  wreck  of  a  Portuguese 
vessel  upon  the  shores  of  Japan  in  1542  led  soon  after  to  the 
efforts  of  traders  and  Jesuit  missionaries  to  effect  an  entry  into 
the  country.  In  1549  Xavier,  Torres,  and  Fernandez  landed  at 
Kagoshima  in  Satsuma.  Since  in  1582  the  Japanese  Christians 
sent  an  embassy  carrying  gifts  to  Rome,  and  since  it  was 
claimed  about  that  time  that  twelve  thousand2  converts  to 
Christianity  had  been  received  into  the  Church,  the  influence 
of  these  missionaries,  and  particularly  that  of  the  "Apostle  of 
the  Indies,"  St.  Francis  Xavier,  must  have  been  great.  In  1587 
the  missionaries  were  ordered  to  be  banished  from  Japan,  and 
during  the  next  forty  years  a  process  of  extermination  of 
Christianity  was  pursued  throughout  the  country. 

In  none  of  this  work,  not  even  in  the  schools  that  the 
Jesuits  are  known  to  have  established  in  Japan,  have  we  a 
definite  trace  of  any  instruction  in  mathematics.  Nevertheless 
the  influence  of  the  most  learned  order  of  priests  that  Europe 
then  produced,  a  priesthood  that  included  in  its  membership 
men  of  marked  ability  in  astronomy  and  pure  mathematics, 
must  have  been  felt.  If  it  merely  suggested  the  nature  of  the 
mathematical  researches  of  the  West  this  would  have  been 
sufficient  to  account  for  some  of  the  renewed  activity  of  the 
seventeenth  century  in  the  scientific  circles  of  Japan.  That 
the  influence  of  the  missionaries  on  mathematics  was  manifested 
in  any  other  way  than  this  there  is  not  the  slightest  evidence. 

It  should  also  be  mentioned  that  an  Englishman  named 
William  Adams  lived  in  Yedo  for  some  time  early  in  the 
seventeenth  century  and  was  at  the  court  of  lyeyasu.  Since 
he  gave  instruction  in  the  art  of  shipbuilding  and  received 
honors  at  court,  his  opportunity  for  influencing  some  of  the 
practical  mathematics  of  the  country  must  be  acknowledged. 
There  is  also  extant  in  a  manuscript,  the  Kikujutsu  Denrai  no 
Maki,  a  story  that  one  Higuchi  Gonyemon  of  Nagasaki,  a 

1  There    is    only    the    merest   mention   of  it   in    P.  HARZER'S   Die  exakten 
Wissenschaften  im  alien  Japan,  Kiel,   1905. 

2  Some    even    claimed    200,000,    at   least    a  little   later.     E.   BOHUM,   Geo- 
graphical Dictionary,  London,   1688. 


256  XIV.  The  Introduction  of  Occidental  Mathematics. 

scholar  of  merit  in  the  field  of  astronomy  and  astrology,  learned 
the  art  of  surveying  from  a  Dutchman  named  Caspar,  and 
not  only  transmitted  this  knowledge  to  his  people  but  also 
constructed  instruments  after  the  style  of  those  used  in  Europe. 
Of  his  life  we  know  nothing  further,  but  a  note  is  added  to 
the  effect  that  he  died  during  the  reign  of  the  third  Shogun 
(1623 — 1650).  A  further  note  in  the  same  manuscript  relates 
that  from  1792  to  1796  a  certain  Dutchman,  one  Peter  Walius(?) 
gave  instruction  in  the  art  of  surveying,  but  of  him  we  know 
nothing  further. 

In  the  eighteenth  century  the  possibility  that  showed  itself 
in  the  seventeenth  century  became  an  actuality.  European 
sciences  now  began  to  penetrate  into  Japanese  schools,  either 
directly  or  through  China.  In  the  year  1713,  for  example,  the 
elaborate  Chinese  treatises,  the  Li-hsiang  K'ao-ch'eng  and  the 
Su-li  Ching-Yun,  which  had  been  compiled  by  Imperial  edict, 
were  published  in  Peking.  Of  these  the  former  was  an 
astronomy  and  the  latter  a  work  on  pure  mathematics,  and 
each  showed  a  good  deal  of  Jesuit  influence.  These  books 
were  early  taken  to  Japan,  and  thus  some  of  the  trend  of 
European  science  came  to  be  known  to  the  scholars  of  that 
country.  There  was  also  sent  across  the  China  Sea  the  Li- 
suan  Ctiiian-shii  in  which  Mei  Wen-ting's  works  were  collected, 
so  that  Japanese  mathematicians  not  only  came  into  some 
contact  with  Europe,  but  also  came  to  see  the  progress  of 
their  science  among  their  powerful  neighbors  of  Asia.  Takebe, 
for  example,  is  said  to  have  studied  Mei's  works  and  to  have 
written  some  monographs  upon  them  in  I726.1 

Nakane  Genkei  (1662 — 1733)  also  wrote,  about  the  same 
time,  a  trigonometry  and  an  astronomy  (see  Fig.  64)  based  on 
the  European  treatment,2  the  result  certainly  of  a  study  of 
Mei  Wen-ting's  works  and  possibly  of  the  Su-li  Cliing-Yun. 


1  ENDO,  Book  II,  p.  69.     There  is  a  copy  in  the  Imperial  Library. 

2  The  Hassen-hyo  Kaigi  (Notes   on   the    Eight    Trigonometric   Lines),    and 
the  Tenmon  Zul^wai  Hakki  (1696).     He  also  wrote  the  Kowa  Tsureki  and  the 
Kb  reki  Sampo  (1714). 


XIV.  The  Introduction  of  Occidental  Mathematics.  257 


6 
> 
£, 


Fig.  64.     From  Nakane  Genkei's  astronomy  of  1696. 

His  pupil  Koda  Shin-yei,  who  died  in  1758,  also  wrote  upon 
the  same  subject.  The  illustrations  given  from  the  works  on 
surveying  by  Ogino  Nobutomo  in  his  Kiku  Genpo  Choken  of 
1718  (Fig.  65),  and  Murai  Masahiro  in  his  Riochi  Shinan  of 

17 


258  XIV.  The  Introduction  of  Occidental  Mathematics. 


Fig.  65.     From  Ogino  Xobutomo's   Kiku  Genpo  Choken  (1718). 


XIV.  The  Introduction  of  Occidental  Mathematics. 


259 


about  the  same  time  (Fig.  66)  show  distinctly  the  European 
influence. 

Later  writers  carried  the  subject  of  trigonometry  still  further. 
For  example,  in  Lord  Arima's  Shnki  Sampd  of  1769  there 
appear  some  problems  in  spherical  trigonometry,  and  in  Sakabe's 
Sanipo  Tenzan  Shinan-roku  of  1810  — 1815  the  work  is  even 
more  advanced.  Manuscripts  of  Ajima  and  Takahashi  upon 
the  same  subject  are  also  extant.  Yegawa  Keishi's  treatise 


Fig.  66.     From  Murai  Masahiro's  Riochi  Shinan. 

on  spherical  trigonometry  appeared  in  1842.  Some  of  the 
illustrations  of  the  manuscripts  on  surveying  are  of  interest,  as 
is  seen  in  the  reproductions  from  Igarashi  Atsuyoshi's  Shinki 
Sokurio  ho  of  about  1775  (Fig.  67)  and  from  a  later  ano- 
nymous work  (Fig.  68). 

The  European  arithmetic  began  to  find  its  way  into  Japan 
in  the  eighteenth  century,  but  it  never  replaced  the  soroban 
by  the  paper  and  pencil,  and  there  is  no  particular  reason 
why  it  should  do  so.  Probably  the  West  is  more  likely  to 

17* 


260 


XIV.  The  Introduction  of  Occidental  Mathematics. 


return  to  some  form  of  mechanical  calculation,  as  evidenced 
in  the  recent  remarkable  advance  in  calculating  machinery, 
than  is  the  Eastern  and  Russian  and  much  cf  the  Arabian 
mercantile  life  to  give  up  entirely  the  abacus.  Napier's  rods, 
however,  appealed  to  the  Japanese  and  Chinese  computers, 
and  books  upon  their  use  were  written  in  Japan.  Arithmetics 
on  the  foreign  plan  were,  however,  published,  Arizawa  Chitei's 
Chusan  Shiki  of  1725  being  an  example.  In  this  work  Arizawa 
speaks  of  the  "Red-bearded  men's  arithmetic,"  the  Japanese  of 


Fig.  67.     From  Igarashi  Atsuyoshrs  Shinki  Sokurio  ho, 

the  period  sometimes  calling  Europeans  by  this  name, — the 
title  Barbarossa  of  the  medieval  West.  Senno's  works  of  1767 
and  1768  were  upon  the  same  subject,  not  to  speak  of  several 
others,  including  Hanai  Kenkichi's  Seisan  Sokuclii  as  late  as 
the  Ansei  (1854 — 1860)  period.  (See  Fig.  69.)  It  is  a  matter 
of  tradition  that  Mayeno  Ryotaku  (1723 — 1803)  received  an 
arithmetic  in  1773  from  some  Dutch  trader,  but  nothing  is 
known  of  the  work.  Mayeno  was  a  physician,  and  in  1769, 
at  the  age  of  forty- six,  he  began  those  linguistic  studies  that 
made  him  well  known  in  his  country.  He  translated  several 
Dutch  works,  including  a  few  on  astronomy,  but  we  have  no 


XIV*.  The  Introduction  of  Occidental  Mathematics.  261 


fr  JSL  %  -f 


f 


t 


A 


ft   "ft   Jg 


Fig.  68.     From  an  anonymous  manuscript  on  surveying. 


262 


XIV.  The  Introduction  of  Occidental  Mathematics. 


evidence  of  his  having  studied  European  mathematics.  Never- 
theless one  cannot  be  in  touch  with  the  scientific  literature  of 
a  language  without  coming  in  contact  with  the  general  trend 


Fig.  69.     From  Hanai  Kenkichr's  Seisan  Sokuchi,  showing 
the  Napier  rods. 

of  thought  in  various  lines,  and  it  is  hardly  possible  that 
Mayeno  failed  to  communicate  to  mathematicians  the  nature 
of  the  work  of  their  unknown  confreres  abroad. 


XIV7.  The  Introduction  of  Occidental  Mathematics.  263 

Contemporary  with  Mayeno  was  scholar  by  the  name  of 
Shizuki  Tadao  (1760— 1806),'  an  interpreter  for  the  merchants 
at  Nagasaki.  At  the  close  of  the  eighteenth  century,  he  began 
a  work  entitled  Rekisho  Shins/id,2  consisting  of  three  parts, 
each  containing  two  books,  the  composition  of  which  was 
completed  in  1803.  The  treatise,  which  was  never  printed, 
is  based  upon  the  works  of  John  Keill.^  The  first  part  treated 
of  the  Copernican  system  of  astronomy  and  the  second  and 
third  parts  of  mechanical  theories.  The  latter  part  of  the 
work  may  have  had  its  inspiration  in  Newton's  Principia.  It 
was  the  first  Japanese  work  to  treat  of  mechanics  and  physics, 
and  it  is  noteworthy  also  from  the  fact  that  the  appendix  to 
the  third  part  contains  a  nebular  hypothesis  that  is  claimed 
to  have  been  independent  of  that  of  Kant  and  Laplace.  Since 
by  the  statement  of  Shizuki  his  theory  dated  in  his  own  mind 
from  about  I/93,4  while  Kant  (1724 — 1804)  had  suggested  it 
as  early  as  1755,  although  Laplace  (1749—1827)  did  not 
publish  his  own  speculations  upon  it  until  1796,5  he  may 
have  received  some  intimation  of  Kant's  theory.  Nevertheless 
Laplace  is  known  to  have  stated  his  theory  independently,  so 
that  Shizuki  may  reasonably  be  thought  to  have  done  the 
same. 

It  should  also  he  stated  that  in  Aida  Ammei's  manuscript 
entitled  Sampo  Densho  Mokuroku  (A  list  of  Mathematical 
Compositions)  mention  is  made  of  an  Oranda  Sampo  (Dutch 
arithmetic).  This  must  have  been  about  1790. 

Contemporary  with  Shizuki  was  the  astronomer  Takahashi 
Shiji,  who  died  in  1804,  aged  forty.  He  was  familiar  with  the 

1  He  is  represented  in  ENDO'S  History,  Book  III,  p.  36,  as  Nakano  Ryuho, 
RyQho  being  his  nom  de  plume,  and  the    date  of  his  book  is  given  as  1797. 

2  New  Treatise  on  subjects  relating  to  the  theory  of  Calendars. 

3  John  Keill  (1671  — 1721),    professor   of  astronomy   at  Oxford.     It  is  said 
by  Dr.  Korteweg  to  have  been  based  upon  a  Dutch  translation  of  these  works; 
but  we  fail  to  find  any  save  the  Latin  editions. 

4  K.  KANO,    On   the  Nebular  Theory  of  Shizuki  Tadao  (in  Japanese),  in  the 
Toyo  Gakugei  Zasshi,  Book  XII,   1895,  pp.  294 — 300. 

5  Exposition  du  Systeme  du  Monde,  Paris   1796. 


264  XIV.  The  Introduction  of  Occidental  Mathematics. 

Dutch  works  upon  his  subject,  and  his  writings  contain  ex- 
tracts from  some  one  by  the  name  of  John  Lilius J  and  from 
various  other  European  scholars. 

The  celebrated  geographer  Ino  Chukei  (1745 — 1821),  whose 
great  survey  of  Japan  has  already  been  mentioned,  was  a 
pupil  of  Takahashi's,  who  translated  La  Lande,  and  thus  came 
to  know  of  the  European  theory  of  his  subject,  which  he  carried 
out  in  his  field  work.  It  might  also  be  said  that  the  shape  of 
the  native  Japanese  instruments  used  by  surveyors  early  in  the 
nineteenth  century  (see  Fig.  70)  were  not  unlike  those  in  use  in 
Europe.  They  were  beautifully  made  and  were  as  accurate  as 
could  be  expected  of  any  instrument  not  bearing  a  telescope. 
It  should  be  added  that  Ino  was  not  the  first  to  use  European 
methods  in  his  surveys,  for  Nagakubo  Sekisui  of  Mito  learned 
the  art  of  map  drawing  from  a  Dutchman  in  Nagasaki,  and 
published  a  map  on  this  plan  in  1789. 

Takahashi  Shiji's  son,  Takahashi  Kageyasu,2  was  also  a 
Shogunate  astronomer  and  as  already  related  he  died  in  prison 
for  having  exchanged  maps  with  a  German  scientist  in  the 
Dutch  service.  This  scientist  was  Philip  Franz  von  Siebold 
(1796  —  1866),  the  first  European  scientist  to  explore  the  country. 
He  was  born  at  Wiirzburg,  Germany,  and  attended  the  uni- 
versity there.  In  1822  he  entered  the  service  of  the  King  of 
the  Netherlands  as  medical  officer  in  the  East  Indian  army, 
and  was  sent  to  Deshima,  the  Dutch  settlement  at  Nagasaki. 
His  medical  skill  enabled  him  to  come  in  contact  with  Japanese 
people  of  all  ranks,  and  in  this  way  he  had  comparatively 
free  access  to  the  interior  of  the  country.  Well  trained  as  a 
scientist  and  well  supplied  with  scientific  instruments  and  with 
a  considerable  number  of  native  collectors,  he  secured  a  large 
amount  of  scientific  information  concerning  a  people  whose 

1  This    is    recorded   in    the    list    of   his   writings    prepared    by    Shibukawa 
Keiyu,  Takahashi's  second  son.     The  name  there  appears  in  Japanese  letters 
as  Ririusu,  with  the  usual  transliteration  of  r  for  /.  Very  likely  it  was  some- 
thing from  the  writings  of  the  well-known   astrologer  William  Lilly. 

2  Also    called    Takahashi   Keiho,   Kageyasu  being  merely  another  reading 
of  Keiho. 


XIV.  The  Introduction  of  Occidental  Mathematics.  265 

customs  and  country  up  to  this  time  had  been  practically 
unknown  to  the  European  world.  As  a  result  he  published 
in  1824  his  De  Historia  Fauna  Japonica,  and  in  1826  his 
Epitome  Lingua  Japonicce.  He  later  published  his  Catalogus 
Librorum  Japonic  orum,  Isagoge  in  Bibliothecam  Japonicam,  and 


Fig.  7°-    Native  Japanese  surveying  instrument.    Early  nineteenth  century. 

Bibliotheca  Japonica,  besides  other  works  on  Japan  and  its 
people.  It  is  thus  apparent  that  by  the  close  of  the  first 
quarter  of  the  nineteenth  century  Japan  was  fairly  well  known 
to  the  outer  world,  and  that  foreign  science  was  influencing 
the  work  of  Japanese  scholars. 


266  XIV.  The  Introduction  of  Occidental  Mathematics. 

Indeed  as  early  as  1811  this  interrelation  of  knowledge  had 
so  far  advanced  that  a  Board  of  Translation  was  established 
in  the  Astronomical  Observatory  in  Yedo,  being  afterward  (1857) 
changed  into  an  Institute  for  the  Investigation  of  European 
Books.  Both  of  these  titles  were  auspicious,  but  they  proved 
disappointing  misnomers.  Not  until  1837  was  any  noteworthy 
result  of  the  labors  of  the  Institute  apparent,  and  then  only 
in  the  preparation  of  the  Seireki  Shimpen  by  Yamaji  Kaiko1 
and  a  few  others,  and  in  a  translation  of  La  Lande.2 

Foreign  influence  shows  itself  indirectly  in  a  manuscript 
written  in  1812  by  Sakabe  Kohan.  This  is  upon  the  theory 
of  navigation  and  is  based  upon  the  spherical  astronomy  of 
the  West.  Another  work  along  the  same  lines,  the  Kairo 
Anshin-roku,  was  published  in  1816  by  Sakabe. 

In  1823  Yoshio  Shunzo  published  his  Yensei  Kansho  Zusetsu, 
in  three  books.  This  work  is  confessedly  based  upon  the 
Dutch  works  of  Martin3  and  Martinet,"1  as  is  stated  in  the 
introductory  note  by  Kusano  Yojun.s 

1  Grandson   of  Yamaji    Shuju,   also    a   Shogunate    astronomer.     The   work 
was  never  printed. 

2  It  is  sometimes  said   that   this  was   based   on  Beima's  works.     But  Elte 
Martens  Beima  (1801  — 1873)  wrote  works  on  the  rings  of  Saturn  that  appeared 
in   1842  and  1843,  and  there  is  no  other  Dutch  writer  of  note  on  astronomy 
by  this  name. 

3  Probably  Martinus  Martens,    Inwijings  Redenvoering  over  eenige  Vborname 
Nuttigheden  der  IVisen  Sterrekunde,  Amsterdam,    1743,  since  Yoshio  speaks  of 
it  as  published  sixty  years  earlier. 

4  Johannes  Florentius  Martinet  (1729 — 1792).    His  Katechisuius  der  Natuur 
(1777 — !779)  is  recorded  by  D.  BIERENS  DE  HAAN  (Bibliographie  Neerlandaise, 
Rome,    1883,   p.  183)   as   having   been   translated   into  Japanese   by   Sammon 
Samme,   but   with   no   information  as   to  publication.     Professor  T.  HAYASHI, 
who   has    given   scholarly    attention   to   this  subject,  is    able  to  find  no  trace 
of    this    translation.     See    his    articles,    A  •  list    of  Dutch    Astronomical   Works 

-imported  from  Holland  to  Japan,  How  have  the  Japanese  used  the  Dutfh  Books 
imported  from  Holland,  and  Some  Dutch  Books  on  Mathematical  and  Physical 
Sciences,  etc.,  in  the  Nieuw  Archie/  voor  Wiskunde,  tweede  reeks,  zevede  deel, 
and  negende  deel.  Possibly  the  translation  was  merely  Yoshio's  work  above 
mentioned,  since  its  secondary  title  is  Catechism  of  Science. 

5  The    work    was    published    by   him    as   having   been   orally   dictated   by 
Yoshio  Shunzo. 


XIV.  The  Introduction  of  Occidental  Mathematics. 


267 


In  the  Tempo  Period  (1830—1844)  Koide  Shuki  translated 
some  portions  of  Lalande's  work  on  astronomy,  and  showed 
to  the  Astronomical  Board  the  superiority  of  the  European 
calendar,  but  without  noticeable  effect.1 

In  1843  Iwata  Seiyo  published  his  Kubo  Shinkei  Shind  (a 
work  relating  to  the  telescope)  in  which  he  made  use  of 
European  methods  in  astronomy.2 


Fig.  71.     Native  Japanese  surveying  instrument. 
Early  nineteenth  century. 

In  1851  Watanabe  Ishin  published  a  work  on  Illustrating  the 
Use  of  the  Octant,  in  which  he  even  adopted  the  Latin  term 
as  appears  by  the  title, —  Okittanto  Yd  ho  Ryakn-zusetsu.  He 
was  followed  by  Murata  Tsunemitsu  in  1853  on  the  use  of  the 
sextant.  An  octant  had  been  brought  from  Europe  in  1780, 

*  FUKUDA,  Sampo  Tamatebako,   1879. 
2  ENDO,  Book  III,  p.  131. 


268  XIV.   The  Introduction  of  Occidental  Mathematics. 

but  had  been  kept  in  the  storehouse  of  the  observatory  because 
no  one  on  the  Shogunate  Astronomical  Board  knew  how  to 
use  it.  Finally  Yamaji  Kaiko  and  a  few  others  worked  with 
it  until  they  understood  it,  and  Watanabe,  who  was  an  expert 
in  gunnery,  wrote  the  work  above  mentioned.  He,  however, 
was  not  aware  of  its  use  in  astronomy,  only  showing  how  it 
might  be  employed  in  measuring  distances  in  surveying.1  The 
sextant  was  imported  somewhat  later  than  the  octant,  but  its 
use  was  not  understood  until  Murata  Tsunemitsu  published 
his  work,  and  even  then  it.  was  employed  only  in  terrestrial 
mensuration.2 

The  Japanese  first  learned  of  logarithms  through  the  Chinese 
work,  the  Su-li  Ching-yiin,  printed  at  Peking  in  1713.  This 
was  not  the  only  Chinese  publication  of  the  subject,  however, 
for  it  is  a  curious  fact  that  no  complete  edition  of  Vlacq's 
tables  ^  appeared  in  Europe  after  his  death,  and  that  the  next 
publication4  thereafter  was  in  Peking  in  1721,5  a  monument  to 
Jesuit  learning.  The  effect  of  these  Chinese  works  was  not 
marked,  however.  Ajima,  who  died  in  1798,  was  one  of  the 
first  Japanese  mathematicians  to  employ  logarithms  in  practical 
calculation,  and  his  manuscript  upon  the  subject  was  used  by 
Kusaka  in  writing  the  Fukyu  Sampo  (1798),  but  the  tables 
were  not  printed.  A  page  from  an  anonymous  table  in  an 
undated  manuscript  entitled  Tai  shin  Rio  su  kwt  giving  the 
logarithms  to  seven  decimal  places  is  shown  in  the  illustration 
(Fig.  72).  The  first  printed  work  to  suggest  the  actual  use 
of  the  tables  was  Book  XII  of  Sakabe's  Sampo  Tenzan  Shinan- 
roku  (Treatise  on  Tenzan  Algebra),  published  in  1810 — 1815. 
Speaking  of  them  he  says:  "Although  these  tables  save  much 
labor,  they  are  but  little  known  for  the  reason  that  they  have 

*  ENDO,  Book  III,  p.  141. 

2  ENDO,  ibid.,  p.  143. 

3  His  Logarithmica  Arithmetica  appeared  at  Gouda  in  1628. 

4  They    had    been    reprinted    in  part  in   GEORGE   MILLER'S  Logarithmicall 
Arilhmetike,  London,  1631. 

5  Magnus  Canon  Logarithmorum  .  .  .  Typis  sinensibus  in  Aula  Pekinensi  jussu 
Imperatoris,  1721. 


XIV.  The  Introduction  of  Occidental  Mathematics. 


269 


never  been  printed  in  our  country.  If  anyone  who  cares  to 
copy  them  will  apply  to  me  I  shall  be  glad  to  lend  them  to 
him  and  to  give  him  detailed  information  as  to  their  use." 
He  gave  the  logarithms  of  the  numbers  I  — 130  to  seven 
decimal  places,  by  way  of  illustration.  He  may  possibly  have 


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,1 


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.— ;\ 

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E.O  o 


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;  o 


JLOO 


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-ttt 

A-tA 


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Fig.  72.     From  an  anonymous  logarithmic  table  in  manuscript. 


2/O  XIV.  The  Introduction  of  Occidental  Mathematics. 

had  some  Dutch  work  on  the  subject,  since  he  knew  the  word 
"logarithm,"  or  possibly  he  had  the  Peking  tables  of  1713 
and  1721. 

Sakabe  further  says:  "The  ratios  involved  in  spherical  triangles, 
as  given  in  the  Li-suan  Ctiuan-shu,  are  so  numerous  that  it 
is  tedious  to  handle  them.  Since  addition  and  subtraction  are 
easier  than  multiplication  and  division,  Europeans  require  their 
calculations  involving  the  eight  trigonometric  lines J  to  be  made 
by  means  of  adding  and  subtracting  logarithms.  They  do  not 
know,  however,  how  to  obtain  the  angles  when  the  three  sides 
are  given,  or  how  to  get  the  sides  from  the  three  angles,2  by 
the  use  of  logarithms  alone." 

The  first  extensive  logarithmic  table  was  printed  by  Koide 
Shuki  (1797 — 1865)  in  1844.  Another  one  was  published  by 
Yegawa  Keishi  in  1857,  in  which  the  logarithms  were  given 
up  to  10,000,3  and  in  the  same  year  an  extensive  table  of 
natural  trigonometric  functions  was  published  by  Okumura  and 
Mori  Masakado,  in  their  Katsu-yen  Hio. 

Although  the  tables  were  used  more  or  less  in  the  first  half 
of  the  nineteenth  century,  the  theory  of  logarithms  remained 
unknown  for  a  long  time  after  it  was  understood  in  China. 
Ajima,  Aida,  Ishiguro,  and  Uchida  Gokan  seem  to  have  been 
the  first  to  pay  any  attention  to  the  nature  of  these  numbers, 
but  few  explanations  were  put  in  print  until  Takemura  Ko- 
kaku  published  his  work  in  1854.  Since  Uchida  used  only 
logarithms  to  the  base  10,  his  theory  as  to  developing  them 
is  very  complicated.4 

It  is  quite  probable  that  some  suggestion  leading  to  the  study 
of  center  of  gravity  found  its  way  in  from  the  West.  Seki 
seems  the  first  to  have  had  the  idea  in  Japan,  and  it  appears  in 
his  investigation  of  the  volume  of  the  solids  generated  by  the 
revolution  of  circular  arcs.  Arima  touches  upon  the  subject 

1  I.  e.,  the   six    common  functions   together  with  the  versed  sine  and  the 
coversed  sine. 

2  Of  a  spherical  triangle. 

3  ENDO,  Book  III,  p.  135. 
*  ENDS,  Book  III,  p.  143. 


XIV.  The  Introduction  of  Occidental  Mathematics.  2/1 

in  the  Shuki  Sampo  of  1769,  and  Takahashi  Shiji  also  mentions 
it.  But  it  was  not  until  after  the  publication  of  Hashimoto's 
work  in  1830,  and  after  there  was  abundant  opportunity  for 
European  influence  to  show  itself,  that  the  problem  became 
at  all  popular.  From  that  time  on  it  was  the  object  of  a 
great  deal  of  attention,  the  solids  becoming  at  times  quite 
complicated.  For  example,  the  center  of  gravity  was  studied 
for  such  a  solid  as  a  segment  of  an  ellipsoid  pierced  by  a 
cylindrical  hole,  and  for  a  group  of  several  circular  cones, 
each  piercing  the  others. 

Similarly  we  may  be  rather  sure  that  the  study  of  various 
roulettes,  including  the  cycloid  and  epicycloid,  came  from  some 
hint  that  these  problems  had  occupied  the  attention  of  mathe- 
maticians in  the  West.  This  does  not  detract  from  the  skill 
shown  by  Wada  Nei,  for  example,  but  it  merely  asserts  that 
the  objects,  not  the  methods  of  study,  were  European  in  source. 
For  the  method,  the  ingenuity,  and  the  patience,  all  credit  is 
due  to  the  Japanese  scholars. 

The  same  remark  may  be  made  with  respect  to  the  catenary 
and  various  other  curves  and  surfaces.  The  catenary  first 
appears  in  Hagiwara's  work  above  mentioned,  and  the  problem 
was  subsequently  solved  by  Omura  Isshii  and  Kagami  Mitsuteru, 
being  attacked  by  approximating,  step  by  step,  the  root  of  a 
transcendental  equation,  a  treatment  very  complicated  but  full 
of  interest.  The  treatment  is  purely  Japanese,  even  though 
the  idea  of  the  problem  itself  may  have  found  its  way  in 
through  Dutch  avenues. 

In  the  nineteenth  century  there  were  a  number  of  scholars 
in  Japan  who  possessed  more  or  less  reading  knowledge  of 
the  Dutch  language.  One  of  these  was  Uchida  Gokan  whose 
name  has  just  been  mentioned  in  connection  with  logarithms. 
He  even  called  his  school  by  the  name  "Maternateka." 1  Of 
him  Tani  Shomo  wrote,  in  the  preface  of  a  work  published 
in  i84O,2  these  appreciative  words:  "Uchida  is  a  profoundly 

1  ENDO,  Book  III,  p.  102. 

2  KEMMOCHI'S  Tan-i  Sampj. 


2/2  XIV.  The  Introduction  of  Occidental  Mathematics. 

learned  man,  and  his  knowledge  is  exceeding  broad.  He  is 
master  even  of  the  'mathematica'  of  the  Western  World."  To 
this  knowledge  his  sole  surviving  pupil,  C.  Kawakita,  has  borne 
witness  in  personal  conversation  with  one  of  the  authors  of 
this  history,  and  N.  Okamoto  still  has  some  of  the  European 
books  formerly  owned  by  Uchida.  Mr.  Kawakita  assures  us, 
however,  that  Uchida's  higher  mathematics  was  his  own 
and  was  not  derived  from  Dutch  sources,  meaning  that  the 
method  of  treatment  was,  as  we  have  already  asserted,  purely 
Japanese. 

In  a  manuscript1  written  in  1824  Ichino  Mokyo  tells  of  an 
ellipsograph  that  Aida  Ammei  designed  from  a  drawing  in 
some  Dutch  work.  "In  reading  some  Occidental  works  recently," 
he  says,  "we  have  seen  recorded  a  method  of  drawing  an 
ellipse  that  is  at  the  same  time  very  simple  and  very  satis- 
factory," and  he  speaks  of  the  fact  that  the  rectification  of 
the  ellipse  by  Japanese  scholars  is  entirely  original  with  them. 
Indeed  it  would  seem  that  the  scholars  of  the  early  nineteenth 
century  were  quite  doubtful  as  to  the  superiority  of  the  European 
mathematics  over  their  own,  which  is  a  rather  unexpected 
testimony  to  the  independence  of  the  Japanese  in  this  science. 
Thus  Oyamada  Yosei  uses  these  words  upon  the  subject:2 
"Mogami  Tokunai  says  in  his  Sokuryo  Sansaku  that  the  mathe- 
matical science  of  our  country  is  unsurpassed  by  that  of  either 
China  or  Europe."  In  the  same  spirit  an  anonymous  writer 
of  the  early  part  of  the  nineteenth  century  writes3  these  words: 
"There  is  an  Occidental  work  wherein  the  value  of  the  circum- 
ference of  a  circle  is  given  to  fifty  figures,  and  of  this  I 
possess  a  translation  which  I  obtained  from  Shibukawa.  It 
is  said  that  this  is  fully  described  by  Montucla  in  his  History 
of  the  Quadrature  of  the  Circle,  published  in  I/54,4  but  I  under- 

1  The  Dayen-shii  Tsujulsu  (General  Method  of  Rectifying  the  Ellipse). 

2  In   the   Malsunoya  Hikki,   an   article   on    Mathematics   and   the   Soroban, 
written  early  in  the  nineteenth  century. 

3  Unpublished  manuscript. 

4  JEAN   ETIENNE   MONTUCLA,   Histoirc    des   recherches  sur   la    quadrature   du 
cercle,  Paris,  1754. 


XIV.  The  Introduction  of  Occidental  Mathematics.  273 

stand  that  this  work  has  not  been  brought  to  Japan.  I,  however, 
have  also  calculated  of  late,  with  the  help  of  Kubodera,  the 
value  to  sixty  figures,  and  not  in  a  single  figure  does  it  differ 
from  the  European  result.  This  goes  to  show  how  exact 
should  be  all  mathematical  work,  and  how,  when  this  accuracy 
is  attained,  the  results  are  the  same  even  though  the  calcul- 
ations be  made  by  men  who  are  thousands  of  miles  apart." 
The  same  writer  also  says:1  "Although  the  Europeans  highly 
excel  in  all  matters  relating  to  astronomy  and  the  calendar, 
nevertheless  their  mathematical  theories  are  inferior  to  those 
that  we  have  so  accurately  developed.  I  one  time  read  the 
Su-li  Ching-yun,  compiled  by  Imperial  edict,  and  in  it  I  found 
a  method  of  solving  a  right  triangle  for  integral  sides,  .  .  .  but 
the  process  was  much  too  cumbersome  and  it  was  lacking  in 
directness.  .  .  .  Moreover  I  have  found  certain  problems  that 
were  incorrectly  solved,  although  I  shall  not  mention  them 
specifically  at  this  time.  From  this  we  may  conclude  that 
foreign  mathematics  is  not  on  so  high  a  plane  as  the  mathe- 
matics of  our  own  country." 

Even  such  a  writer  as  Koide  Shuki  had  a  similarly  low 
estimate  of  the  mathematics  of  the  West,  for  he  expressed 
himself  in  these  words:8  "Number  dwells  in  the  heavens  and 
in  the  earth,  but  the  arts  are  of  human  make,  some  being 
accurate  and  others  not.  The  minuteness  of  our  mathematical 
work  far  surpasses  that  to  be  found  in  the  West,  because  our 
power  is  a  divine  inheritance,  fostered  by  the  noble  and  daring 
spirit  of  a  nation  that  is  exalted  over  the  other  nations  of 
the  world." 

In  similar  spirit,  the  lordly  spirit  of  the  old  samurai,  Takaku 
Kenjiro  (1821 — 1883)  writes  in  his  General  View  of  Japanese 
Mathematics: ^  "Astronomy  and  the  physical  sciences  as  found 
in  the  West  are  truth  eternal  and  unchangeable,  and  this  we 
must  learn;  but  as  to  mathematics,  there  Japan  is  leader  of 
the  world." 

1  In  his  Sanwa  Zuihitsii. 

2  In  his  preface  to  KEMMOCHI'S  Tan-i  Sampo,  1840. 

3  For  this  we  are  indebted  to  the  writings  of  C.  KAWAKITA. 

18 


2/4  XIV.  The  Introduction  of  Occidental  Mathematics. 

Hagiwara  Teisuke  (1828 — 1909),  one  of  the  last  of  the 
native  school,  also  bemoaned  the  sacrifice  of  the  wasan  that 
followed  on  the  inroads  of  Western  science.  Of  his  own 
published  problems  he  was  wont  to  say  that  no  European 
mathematician  could  ever  have  solved  them  because  of  their 
very  complicated  nature. 

Such  testimony  may  be  looked  upon  by  some  as  a  display 
of  pitiful  ignorance,  as  in  certain  respects  it  was.  But  on  the 
other  hand  it  bears  testimony  to  the  fact  that  the  mathe- 
maticians of  the  old  school  were  not  looking  to  Europe  for 
assistance,  feeling  rather  that  Europe  should  look  to  them. 

In  view  of  these  opinions  it  is  of  interest  to  read  the  words 
of  a  serious  observer  of  things  Japanese  in  the  seventeenth 
century.  Engelbert  Kaempfer  living  in  Japan  during  the  rule 
of  the  fifth  of  the  Tokugawa  Shoguns  (1680 — 1709)  remarked 
"They  know  nothing  of  mathematics,  especially  of  their  pro- 
found and  speculative  parts.  No  one  interests  himself  in  this 
science  as  we  Europeans  do."1 

The  differential  and  integral  calculus,  in  its  definite  Western 
form,  entered  Japan  through  a  Chinese  version  of  the  American 
work  by  Loomis.2  This  version,  entitled  Tai-wei-chi  Shih-cJd, 
was  translated  by  Li  Shan-Ian  in  1859,  with  the  help  of  Alexander 
Wylie,  an  English  missionary.  About  the  same  time  several 
other  treatises  were  translated  into  Chinese,  but  how  many  of 
these  found  their  way  into  Japan  we  do  not  know. 

As  to  arithmetic  some  mention  has  already  been  made  of 
the  European  influence.  Yamamoto  Hokuzan  says,  in  his 
preface  to  Ohara  Rimei's  Tenzan-Shinan  of  1810,  that  the 
tenzan  algebra  of  the  Seki  school  was  merely  founded  on  the 
European  method  of  computing.  For  this  statement  there 

1  KAEMPFER'S  work  was  translated  from  the  German  by  SCHEUCHZER,  and 
published  in  London  in  1727 — 1728.    This  extract  comes  through  a  German 
retranslation  from  the  English,  by  P.  HARZER,  loc.  cit.,  p.  17. 

2  Elias  Loomis  (1811 — 1899).    Since  the  work  is  on  both  algebra  and  the 
calculus  it  was  probably  compiled  from  the  Elements  of  Algebra,   New  York, 
1846.    and    the    Elements    of  Analytical    Geometry    and  of  the  Differential  and 
Integral  Calculus,  New  York,   1850. 


XIV.  The  Introduction  of  Occidental  Mathematics. 


275 


seems  to  be  no  basis,  but  it  shows  that  even  in  the  nineteenth 
century  the  Western  methods  of  computation  were  not  at  all 
well  known. 

About   the   middle    of  the   century   the  European   methods 
began  to   find  definite  place  in  Japanese  works,  if  not  in  the 


- 


Xffl 


XHH 
XIV 


XI 


XII 


IX 


-r 

10 


X 


7 


VII 


Mil 


-£>    ^ 


r* 


V 


VI 


III 


TEE? 


mr 


II 


- 

T 


4^ 

^ 


^ 


Fig.  73.     From  Hanai  Kenkichi's 


Sokuchi  (1856). 


schools.  The  first  of  these  works  was  Hanai  Kenkichi's 
Seisan  Sokuchi  (Short  Course  in  Western  Arithmetic),  published 
in  1856  (Fig.  73),  and  Yanagawa  Shunzo's  Yosan  Yoho  (Methods 


18* 


2/6  XIV.    The  Introduction  of  Occidental  Mathematics. 

of  Western  Arithmetic)  that  appeared  in  the  same  year.  The 
influence  of  these  and  similar  books  of  later  date  has  been 
on  pedagogical  and  commercial  rather  than  on  mathematical 
lines.  The  soroban  is  as  popular  as  ever,  and  save  for  those 
who  proceed  to  higher  mathematics  it  seems  destined  to  re- 
main so. 

It  was  about  the  year  1851  that  the  Shogunate  ordered 
certain  persons  to  be  instructed  by  Dutch  masters  at  Naga- 
saki in  the  art  of  navigation.  As  a  basis  for  this  instruction 
Dutch  arithmetic  was  taught  and  this  seems  to  have  been  the 
first  systematic  instruction  in  the  subject  in  Japan.  In  1855 
an  institute  was  founded  in  Yedo  for  the  same  purpose,  Dutch 
teachers  being  employed.  One  of  the  pupils  in  this  school 
was  Ono  Tomogoro,  and  from  him  we  know  of  the  work  there 
given.1  The  course  extended  over  four  or  five  years,  and 
Li's  version  of  the  work  of  Loomis  was  used  as  a  text- 
book.2 

The  influence  of  such  a  work  as  that  of  Loomis  was  very 
slight,  however.  Scholars  who  knew  European  mathematics 
were  few,  and  the  subject  was  generally  looked  upon  as  of 
inferior  merit.  It  was  not  until  a  generation  later  that  the 
Western  calculus  attracted  much  attention.  Some  of  the 
efforts  at  combining  Eastern  and  Western  mathematics  at  about 
this  period  are  interesting,  as  witness  an  undated  manuscript 
by  one  Wake  Yukimasa,  of  which  a  page  is  here  shown 

(Fig.  74). 

There  exists  in  the  library  of  one  of  the  authors  a  manuscript 
translation  from  the  Dutch  of  Jacob  Floryn  (1751  — 1818), 
entitled  Shinyakuho  Sankaku  Jutsu  (Newly  translated  art  of 
trigonometry).  It  was  made  in  the  Ansei  period  (1854 — 1860) 
by  Takahashtri  Yoshiyasu,  probably  a  member  of  the  family 
of  well-known  mathematicians.  It  is  possibly  from  Floryn's 

1  The    Use   of  Japanese    Mathematics   (in   Japanese)    in    the  Sugaku  Hochi, 
no.  88. 

2  Mr.  K.  UYENO   informs  us    that  the  Loomis  book  was  brought  to  Japan 
before  Li's  translation  was  made,  but  that  there  was  no  one  who  knew  both 
English  and  mathematics  well  enough  to  read  it. 


XIV.  The  Introduction  of  Occidental  Mathematics.  277 

Grondbeginzels  der  Hoogere  Meetkunde  which  was  published 
in  Rotterdam  in  1794.  This  translation  seems  not  to  be  known. 
Of  the  conic  sections  some  intimation  of  the  subject  may 
have  reached  Japan  in  the  seventeenth  century,  but  it  evi- 
dently was  taken,  if  at  all,  only  as  a  hint.  The  Japanese 
studied  the  ellipse  very  zealously,  always  by  their  own  peculiar 


JiL  =  Q 


:  -f- ==  -i-  /  © 


Fig.  74.     From  a  manuscript  by  Wake  Yukismasa. 

method,  but  the  parabola  and  hyperbola  seem  never  to  have 
attracted  the  attention  of  the  old  school.  To  be  sure,  the 
parabola  enters  into  a  problem  about  the  path  of  a  projectile 
in  Yamada's  Kaisanki  of  1656,  but  it  seems  never  to  have 
been  noticed  by  subsequent  writers.  The  graphs  of  these 
curves  are  found  in  certain  astronomical  works,  as  in  Yoshio's 
Yensei  Kansho  Zusetsu  of  1823  where  they  are  used  in  illustrat- 


2/8  XIV.  The  Introduction  of  Occidental  Mathematics. 

ing  the  orbits  of  comets,  but  they  do  not  enter  into  the  works 
on  pure  mathematics.  This  very  fact  is  evidence  against  any 
influence  from  without  affecting  the  native  theories. 

We  have  already  spoken  of  the  change  of  the  Board  of 
Translation  to  the  Institute  for  the  Investigation  of  European 
Books.  Six  years  after  this  change  was  made  the  Kaiseijo 
School  was  founded  (1863),  in  which  every  art  and  science 
was  to  be  taught.  A  department  of  mathematics  was  included, 
and  in  this  Kanda  Kohei  was  made  professor.  He  it  was  who 
made  the  first  decisive  step  towards  the  teaching  of  European 
mathematics  in  Japan,  and  from  his  time  on  the  subject  re- 
ceived earnest  attention  in  spite  of  the  small  number  of  students 
in  the  department. 

The  year  1868  is  well  known  in  the  West  and  in  Japan 
as  a  year  of  great  import  to  the  world.  This  was  the  year 
of  the  political  revolution  that  overthrew  the  Tokugawa  Sho- 
gunate,  that  put  an  end  to  the  feudal  order,  and  that  restored 
the  Imperial  administration.  Yedo,  the  Shogun's  capital,  became 
Tokyo,  the  seat  of  the  Empire.  The  year  is  known  to  the 
West  because  it  marked  the  coming  of  a  new  World  Power. 
What  this  has  meant  the  past  forty  years  have  shown;  what 
it  is  to  mean  as  the  centuries  go  on,  no  one  has  the  slightest 
conception.  To  Japan  the  year  marks  the  entrance  of  Western 
ideas,  many  of  which  are  good,  and  many  of  which  have  been 
harmful.  The  art  of  Japan  has  suffered,  in  painting,  in  sculp- 
ture, and  especially  in  architecture.  The  exquisite  taste  of  a 
century  ago,  in  textiles  for  example,  has  given  place  to  a  catering 
to  the  bad  taste  of  moneyed  tourists.  And  all  of  this  has  its 
parallel  in  the  domain  of  mathematics,  in  which  domain  we 
may  now  take  a  retrospective  view. 

What  of  the  native  mathematics  of  Japan,  and  what  of  the 
effect  of  the  new  mathematics?  What  did  Japan  originate  and 
what  did  she  borrow?  What  was  the  status  of  the  subject 
before  the  year  1868,  and  what  is  its  status  at  the  present 
and  its  promise  for  the  future? 

Looked  at  from  the  standpoint  of  the  West,  and  weighing 
the  evidence  as  carefully  and  as  impartially  as  human  imper- 


XIV.  The  Introduction  of  Occidental  Mathematics.  279 

factions   will    allow,    this    seems   to   be    a    fair  estimate  of  the 
ancient  wasan: — 

The  Japanese,  beginning  in  the  seventeenth  century,  produc- 
ed a  succession  of  worthy  mathematicians.  Since  these  men 
studied  the  general  lines  that  interested  European  scholars  of 
a  generation  earlier,  and  since  there  was  some  opportunity 
for  knowing  of  these  lines  of  Western  interest,  it  seems  reason- 
able to  suppose  that  they  had  some  hint  of  what  was  occu- 
pying the  attention  of  investigators  abroad.  Since  their  me- 
thods of  treatment  of  every  subject  were  peculiar  to  Japan, 
either  her  scholars  did  not  value  or,  what  is  quite  certain, 
did  not  know  the  detailed  methods  of  the  West.  Since  they 
decried  the  European  learning  in  mathematics,  it  is  probable 
that  they  made  no  effort  to  know  in  detail  what  was  being 
done  by  the  scholars  of  Holland  and  France,  of  England  and 
Germany,  of  Italy  and  Switzerland. 

With  such  intimation  as  they  may  have  had  respecting  the 
lines  of  research  in  the  West,  Japan  developed  a  system  of 
her  own  for  the  use  of  infinite  series  in  the  work  of  mensur- 
ation. She  later  developed  an  integral  calculus  that  was 
sufficient  for  the  purposes  of  measuring  the  circle,  sphere, 
and  ellipse.  In  the  solution  of  higher  numerical  equations  she 
improved  upon  the  work  of  those  Chinese  scholars  who  had 
long  anticipated  Horner's  method  in  England.  In  the  study 
of  conies  her  scholars  paid  much  attention  to  the  ellipse  but 
none  to  the  parabola  and  hyperbola. 

But  the  mathematics  of  Japan  was  like  her  art,  exquisite 
rather  than  grand.  She  never  develpoed  a  great  theory  that 
in  any  way  compares  with  the  calculus  as  it  existed  when 
Cauchy,  for  example,  had  finished  with  it.  When  we  think 
of  Descartes's  La  Geometric;  of  Desargues's  Brouillon  proiect, 
of  the  work  of  Newton  and  Leibnitz  on  the  calculus;  of  that 
of  Euler  on  the  imaginary,  for  example;  of  Lagrange  and 
Gauss  in  relation  to  the  theory  of  numbers;  of  Galois  in  the  dis- 
covery of  groups,  —  and  so  on  through  a  long  array  of  names, 
we  do  not  find  work  of  this  kind  being  done  in  Japan,  nor  have 
we  the  slightest  reason  for  thinking  that  we  ought  to  find  it. 


28O  XIV.    The  Introduction  of  Occidental  Mathematics. 

Europe  had  several  thousand  years  of  mathematics  back  of 
her  when  Newton  and  Leibnitz  worked  on  the  calculus, — 
years  in  which  every  nation  knew  or  might  know  what  its 
neighbors  were  doing;  years  in  which  the  scholars  of  one 
country  inspired  those  of  another.  Japan  had  had  hardly  a 
century  of  real  opportunity  in  mathematics  when  Seki  entered 
the  field.  From  the  standard  of  opportunity  Japan  did  remark- 
able work;  from  the  standpoint  of  mathematical  discovery  this 
work  was  in  every  way  inferior  to  that  of  the  West. 

When,  however,  we  come  to  execution  it  is  like  picking  up 
a  box  of  the  real  old  red  lacquer, — not  the  kind  made 
for  sale  in  our  day.  In  execution  the  work  was  exquisite  in 
a  way  wholly  unknown  in  the  West.  For  patience,  for  the 
everlasting  taking  of  pains,  for  ingenuity  in  untangling  minute 
knots  and  thousands  of  them,  the  problem-solving  of  the  Ja- 
panese and  the  working  out  of  some  of  the  series  in  the  yenri 
have  never  been  equaled. 

And  what  will  be  the  result  of  the  introduction  of  the  new 
mathematics  into  Japan?  It  is  altogether  too  early  to  foresee, 
just  as  we  cannot  foresee  the  effect  of  the  introduction  of 
new  art,  of  new  standards  of  living,  of  machinery,  and  of  all 
that  goes  to  make  the  New  Japan.  If  it  shall  lead  to  the 
application  of  the  peculiar  genius  of  the  old  school,  the  genius 
for  taking  infinite  pains,  to  large  questions  in  mathematics, 
then  the  world  may  see  results  that  will  be  epoch  making. 
If  on  the  other  hand  it  shall  lead  to  a  contempt  for  the  past, 
and  to  a  desire  to  abandon  the  very  thing  that  makes  the 
wasan  worthy  of  study,  then  we  cannot  see  what  the  future 
may  have  in  store.  It  is  in  the  hope  that  the  West  may 
appreciate  the  peculiar  genius  that  shows  itself  in  the  works 
of  men  like  Seki,  Takebe,  Ajima,  and  Wada,  and  may  be  sym- 
pathetic with  the  application  of  that  genius  to  the  new  mathe- 
matics of  Japan,  that  this  work  is  written. 


INDEX 


Abo  no  Seimei  67. 

Adams  255. 

Ahmes  papyrus  51,   104. 

Aida  Ammei  172,  177,  188,  193, 

263,  272. 
Ajima  Chokuyen  163,   191,   195, 

218,  220,  221,  224,  259,  268. 
Akita  Yoshiichi  219. 
Algebra  49,   50,   105. 

See  Equations. 
Algebra,  name   104. 
Almans   137. 
Ando  Kichiji  130. 
An  do  Yuyeki  63. 
Andrews  69. 
Aoyama  Riyei  77,   164. 
Apianus   114. 
Araki  Hikoshiro   Sonyei  33,  45, 

104,   107,   155,   158. 
Arima    Raido     106,     161,    181, 

182,   186,   197,  208,  259,  270. 
Arizawa  Chitei  260. 
As£,da  Goryu  141,   206,   207. 
Astronomy  263. 

Baba  Nobutake  166. 
Baba  Seito  248. 
Baba  Seitoku  172. 
Baisho  52. 


Baker  197. 

Bamboo  rods  21,  23,  47. 

Ban  Seiyei  197,  198. 

Bernoulli  197. 

Bierens  de  Haan  266. 

Biernatzki  12. 

Binomial  theorem  51,  182,   193. 

Bohlen,  von  6. 

Bohum  255. 

Bostow  137. 

Bowring  30. 

Buddhism  7,   15,   17. 

Bushido   1 4. 

Calculus  87,   123,   272. 

See  Yenri. 
Cantor  133. 
Carron  135,  138. 
Carus  iv,    20. 
Caspar  256. 
Casting  out  nines   170. 
Catenary  248. 
Cauchy  126. 

Cavalieri  85,  86,   123,  157,  162. 
Celestial  element  49,  52,  77,  86, 

132. 

Celestial  monad  50. 
Center  of  gravity  217,  242,  270. 
Chang  Heng  63. 


282 


Index. 


Chang  T'sang  48. 

Chen  49. 

Ch'en  Huo  63. 

Cheng  and  fu  48. 

Ch'eng  Tai-wei  34. 

Chiang  Chou  Li  Wend  49. 

Chiba  Saiyin   171. 

Chiba  Tanehide  244. 

Ch'in  Chiu-shao  48,  49,   50,  63. 

China  i,  9,  48,   57. 

Chinese   works   9,   33,   35,    in, 

115,  129,  132,  146,  168,  213, 

254,  256,  268. 
Chiu-chang  9,   n,  48. 
Chiu-szu  9,   ii. 
Chou-pei  Suan-ching  9. 
Chu  Chi-chieh  48,  49,  51,  52,  56. 
Chui-shu  9,   14. 
Circle  60,  63,  76,  77,   109,  131. 

See  TT. 

Colebrooke  5. 

Continued  fractions  145,  200. 
Counting  4. 
Courant  22. 
Cramer  126. 
Cycloid  248. 


De  la  Couperie  18. 
Descartes   133. 
Determinants   124. 
Differences,  method  of  106,  107, 

148,  234. 
Diophantine  analysis   196. 

See  Indeterminate  equations. 
Di  san  Filipo  6. 
Dowun  52. 


Dutch   influence    132,   136,   140, 

206,  217,  254,  256,  260,  263, 
271,  272,  276. 

Ellipse  69,  206. 

Elliptic  wedges  250. 

Endo  iv,  4,  9,  15,  17,  33,  35, 
60  —  63,  65,  78,  79,  85,  91  — 
93,  95,  102,  104—106,  123, 
129,  130,  141,  144,  151,  152, 

155.     156,     159,     172,     i77, 
179—181,     197,     200,     204, 

207,  216—225,     227,     243, 
256,  263,  267,  268,  270,  271. 

Epicycloid  248. 

Equations  49,  52,  86,  102,  106, 
113,  129,  138,  168,  172,  182, 

2I2;     213,    224,    225,   226,    229, 
235.     271,     272,     279. 

Euclid  25^ 
Euler  193. 

Fan  problems  231. 

Fernandez  255. 

Floryn   276. 

Folding  process  125. 

Folding  tables   221,  248. 

Fractions  105,   145,   176,   198. 

Fujikawa  136. 

Fujioka  241. 

Fujisawa  iii,   92. 

Fujita  Kagen  184. 

Fujita   Sadasuke    92,    183,    184, 

188,  195. 

Fujita  Seishin  212. 
Fujiwara  Norikaze  46. 


Index. 


283 


Fukuda  Riken  32,   85,  92,   155, 

177,  251,  267. 
Fukuda  Sen   199. 
Fukudai  problems   124. 
Furukawa  Ken  76. 
Furukawa  Ujikiyo   157,   207. 

Genko  60. 
Gensho   17. 
Gentetsu  52. 
Geometry  216,  218. 
Gokai  Amp  on   233,  243. 
Goschkewitsch   18. 
Gow  5. 

Hachiya  Kojuro  Teisho    153. 
Hagiwara  Teisuke  157,  159,  240, 

248,   271,   274. 
Hai-tao  Suan-shu  9,   n. 
Hanai  Kenkichi  260,  275. 
Hartsingius   133,   138. 
Harzer  133,  154,  155,  195,  223, 

255- 

Hasegawa  Ko  248. 
Hashimoto  Shoho   216,  271. 
Hasu  Shigeru   177. 
Hatono  Soha  136,   138. 
Hatsusaka  64. 
Hayashi  in,   18,  23,   26,  33,  65, 

85,  9i,  92,  95,  I07,  H4, 
124,  126,  133,  141,  152,  155, 
159,  193,  200,  266. 

Hayashi  Kichizaemon  140. 

Hazama  Jufu  206,  207. 

Hazama  Jushin  206. 

Hendai  problems  115. 


Higher  equations  50,  52,  86,  93. 

Higuchi  Gonyemon  255. 

Hirauchi  Teishin  215,    219. 

Hitomi  192. 

Hodqji  Wajuro   250. 

Honda    Rimei    143,     172,     188, 

208. 

Honda  Teiken  183. 
Hori-ike  Hisamichi  238. 
Horiye   177. 
Homer's    method    51,    56,    115, 

213. 

Hoshino  Sanenobu  57,   128. 
Hosoi  Kotaku   166. 
Hozumi  Yoshin  163. 
Hsu  Kuang-crfijfijjf"  254. 
Hiibner  18. 

Ichikawa  Danjyuro   166. 

Ichimo  Mokyo   272. 

Idai  Shoto  62. 

Igarashi  Atsuyoshi  259. 

Ikeda  Shoi  129,   130,  235. 

Iku-ko   172. 

Imaginaries  209. 

Imai  Kentei  166,  171. 

Imamura  Chisho  62,  63. 

Indeterminate  equations  168,  182, 

192,   196,  233,   246. 
Infinitesimal  analysis   197. 
Ino  Chukei  206,  264. 
Integration    123,   129,  202,  204, 

221. 

Iriye  Shukei  164,  171. 
Ishigami   136. 
Ishigaya  Shoyeki  144. 
Ishiguro  Shin-yu  62,   231. 


284 


Index. 


Isomaru  Kittoku  17,  45,  62,  64, 

65,   103,   129,   149,   158. 
It  6  Jinsai  166. 
Iwai  Juyen  229,  238. 
Iwasaki  Toshihisa  238,   247. 
Iwata  Kosan  250. 
Iwata  Seiyo  247,  267. 
lyezaki  Zenshi  230. 


Jartoux   154. 

Jesuits  57,  132,    154,  168,  254, 

255,  256- 
Jindai  monji  3. 


Kaempfer  274. 
Kaetzu  247,  250. 
Kagami  Mitsuteru  248,   271. 
Kaiko  266. 
Kakudo  Shoku  244. 
Kamiya  Hotei   166. 
Kamiya  Kokichi  Teirei  189. 
Kamizawa  Teikan  92  —  94. 
Kanda  Kohei  278. 
Kano   62,   166,   192,   263. 
Kanroku  8. 
Kant  263. 

Karpinski  6,  30,  35. 
Katsujutsu  method  123. 
Kawai  Kyutoku   213. 
Kawakita    33,    59,    60,    62,    91, 
146,   155,   159,  177,  183,  188, 

191,     196,    219,    221,    241,   272, 

273- 

Keill  263. 
Keishi-zan  17. 


Kemmochi  Yoschichi  Shoko  228, 

238,   241,  271,  273. 
Kieou-fong  20. 
Kigen  seiho  method  104. 
Kikuchi,  Baron  iii. 
Kikuchi  Choryo  246. 
Kimura  Shoju  237. 
Kimura,  T.,  6. 
Klingsmill  6. 

Knott  4,   1 8,  31,  36,  37,  40. 
Kobayashi  247. 
Kobayashi  Koshin  186. 
Kobayashi  Tadayoshi  242. 
Kobayashi  Yoshinobu  140. 
Kobo  Daishi  15. 
Koda  Shin-yei  170,  257. 
Koide  Shuki  199,  220,  221,  267, 

270,  273. 
Koike  Yuken   172. 
Koko   59. 

Korea  i,  21,  31,  48. 
Kouo  Sheou-kin  21. 
Koyama  Naoaki  220. 
Kozaka  Sadanao   129. 
Kubodera  273. 
Kuichi  Sanjin  92. 
Kuichi  school  129. 
Kuo  Shou-ching  108. 
Kuru  Juson   153. 
Kurushima  Kinai  166,   170. 
Kurushima  Yoshita  176, 179,  181. 
Kusaka  Sei  172,  218,  220,   240, 

268. 

Kusano  Yojun  266. 
Kuwamoto  Masaaki  250. 
Kwaida  Yasuaki  238. 
Kyodai  problems  115. 


Index. 


285 


Lao-tze  20. 
Laplace  263. 
Legge   12. 

Leibnitz  125,  126,  154. 
Leyden  133. 
Li  Shan-Ian  274. 
Li  Show  12. 
Li  Te-Tsi  49. 
Li  Yeh  48—50. 
Lieou  Yi-K'ing  20. 
Lilius  264. 
Liu-chang  9,   10. 
Liu  Hui  48,  63. 
Liu  Ju  Hsieh  49. 
Liu  Ta-Chien  49. 
Lo  Shih-lin  48. 
Locke  v. 
Logarithms  268. 
Loomis  274,  276. 
Lowell  21. 

Magic  circles  71,  79,   120. 
Magic     squares     57,     69,     116, 

177. 

Magic  wheels  73. 
Malfatti  problem  196. 
Mamiya  Rinzo   172. 
Man-o  Tokiharu  163. 
Mathematical    schools    of  Japan 

207,  271. 

Mathematics,  first  printed  61. 
Martin  266. 
Martinet  266. 
Matsuki  Jiroyemon   166. 
Matsunaga   104,    158,   160,   180. 
Matsuoka  180,  231. 
Matteo  Ricci  132,  254. 


Maxima   and   minima    107,   182 

229,  250. 

Mayeno  Ryotaku  141,   260. 
Mechanics  263. 
Mei  Ku-cheng  155. 
Mei  Wen-ting  19,  29,  168,  256. 
Meijin   196. 
Michinori  17. 
Michizane  15. 
Mikami  14,  29,  49,  63,  91,  133, 

138,   144,   147. 
Mtnami  Ryoho   247. 
Mitsuyoshi  59. 

Miyagi  Seiko   129,  130,   179. 
Miyajima  Sonobei  Keichi  185. 
Miyake    Kenryu    27,     46,     83, 

164. 

Mizoguchi  216. 
Mochinaga  129. 

Mogami  Tokunai  143,   172,  272. 
Mohammed  ibn  Musa  104. 
Mohl  20. 

Momokawa  Chubei  43. 
Monbu  9. 
Montucla  272. 
Mori  Kambei  Shigeyoshi  32,  35, 

58,  60,  61,   103. 
Mori  Misaburo  35. 
Mori  Masakado  270. 
Muir  124,   125. 

Murai  Chuzen  15,  34,   172,   174. 
Murai  Mashahiro   164,  257. 
Muramatsu  61,  64,  77,   109. 
Murase   128. 

Murata  Koryu  172,  216. 
Murata    Tsunemitsu    243,    267, 

268. 


286 


Index. 


Murata  Tsushin  45. 
Murray  9. 

Nagakubo  Sekisui  141. 

Nagano  Seiyo   172. 

Naito  Masaki  104,   159. 

Nakamura  64. 

Nakane   Genkei   130,   146,   i66; 

256. 
Nakane  Genjun    164,    166,    169, 

172,   174,   181,   198. 
Nakanishi  Seiko   129. 
Nakanishi  Seiri  129,   188. 
Nakashima  Chozaburo   136. 
Napier's  rods  260. 
Nashimoto   166. 
Nebular  theory  263. 
Newton   115,  193,  263. 
Nines,  check  of  170. 
Nishikawa  Joken   141. 
Nishimura  Yenri  198. 
Nishiwaki  Richyu  27,   163. 
Nitobe   14. 
Nozawa  Teicho  65,    80,  84,  86. 

Oba  Keimei  172. 

Ogino  Nobutomo  164,  257. 

Ogyu  Sorai  166. 

Ohara  Rimei  208,  274. 

Ohashi  129. 

Okamoto  34,  35,  155,  157,  160, 

221,     272. 

Okuda  Yuyeki  128. 
Omura  Isshu  245,  248,  271. 
Otaka  45,  107,  108,   113,   147. 
Oyama  Shokei  152,  156. 
Oyamada  Yosei  31,  143,  272. 


Ozawa   Seiyo   65,    91,    92,    104, 
172. 

Pan  Ku  20. 

Pascal's  triangle   51,   114. 
Pentagonal  star  67. 
Physics  263. 

TT  60,  63,  65,  78,  85,  in,  129, 
144,   i45>  IS1-  J53>  160,  179, 

182,     191,    212,    223,    224. 

Positive  and  negative  48. 

Postow  137. 

Power  series   108. 

Prismatoid   164. 

Pythagorean  theorem  10,  13,  180. 

Rabbi  ben  Ezra  84. 

Recurring  fractions   176,   198. 

Regis  154. 

Regula  falsi  13. 

Regular  polygons   63,   65,    107, 

161. 

Reinaud  6. 
Ricci   132,  254. 
Riken   199. 
Rodet  1 8. 
Roots  212. 

See  Equations,  Square  root, 

Cube  root. 
Roulettes  242,  247,  250. 

Saito  Gicho  242. 

Saito  Gigi  242. 

Sakabe  Kohan    172,   208,   259, 

266,  268,  270. 

Sangi    1 8,    21,    23,  47,  52,  213. 
San-k'ai  Chung-ch'a  9. 


Index. 


28; 


Sato  Moshun  (Shigeharu)  24,  45, 

65,  86,  88,  89,   130. 
Sato  Seiko  85,   130. 
Sawaguchi  Kazuyuki  45,  86,  95, 

130. 

Schambergen  137. 
Schools  207,  271. 
Schotel  135. 
Seki    Kowa    17,    82,    91,    138; 

144,  145,  147,  151,  156,  159, 

209,  218,  225,  270. 
Senno  260. 
Series   161,   177,  200,  203,  211, 

225. 

Sharp   14. 
Shibamura  64. 
Shibukawa  Keiyu  264. 
Shibukawa  Shunkai   130. 
Shih  Hsing  Dao  49. 
Shino  Chikyo   241. 
Shiono  Koteki   144. 
Shiraishi  Chochu  34,  201,    233, 
Shizuki  Tadao   141,  263. 
Shotoku  Taishi  8. 
Siebold  217,  264. 
Skew  surface  242. 
Smith    6,    19,   30,   35,    51,    114, 

124. 

Someya  Harufusa  144. 
Soroban    18,   31,   47,   176,  213, 

259,   276. 
Sou  Lin  20. 
Sphere  63,  76. 
Spiral  163. 

Square  root  176,   177,  200. 
Suan-hsiao  Chi-meng  146. 
Sumner  7,  8. 


Sun-tsu  21. 

Sun-tsu  Suan-ching  9,   10. 

Surveying  256. 

Suzuki  Yen  252. 

Swan-king  9. 

Swan-pan  19,   29,  47. 

T'ai  tsou  29. 

Takebe  Kenko   48,   52,  76,  95, 

98,   103,   104,   112,   128,  129, 

143—146,     151,     153,     158, 

166,  168. 

Takahara  Kisshu  64,  86,  92. 
Takahashi      Kageyasu      (Keiho) 

264. 
Takahashi   Shiji    141,    206,    207, 

217,  259,   263,  264,  271. 
Takahashi  Yoshiyasu  276. 
Takaku  Kenjiro  250,  273. 
Takeda  Saisei  166. 
Takeda  Shingen  216. 
Takeda  Tokunoshin  231. 
Takemura  Kokaku  270. 
Takenouchi  241. 
Takuma  Genzayemon  179. 
Tani  Shomo  271. 
Tanimoto   15. 
Tatamu  process   125. 
Tawara  Kamei  64. 
Tendai  problems   114. 
Tengen  jutsu  48,  102. 
Tenji  9. 
Tenjin  15. 
Tenzan  method    103,   104,  107, 

159,   182,  196,  208,  218,  219, 

243,  274. 
Terauchi  Gompei  159. 


288 


Index. 


Tetsu-jutsu  method  106. 
Tokuhisa  Komatsu  129. 
Torres  255. 
Toyota  243. 
Toyota  Bunkei  182. 
Trapezium  226,    227. 
Trigonometry     196,     213,    25  6, 

259,  276. 
Ts'ai  Ch'en  20. 
Tschotu  19. 

Tsu  Ch'ung-chih  112,   147. 
Tsuboi  Yoshitomo   166. 
Tsu  da  Yenkyu   198. 

Uchida    Gokan     15,     33,    238, 

241,   270,  271. 
Uchida  Kyumei  248. 
Unknown  quantity  51. 
Uyeno   276. 

Van  Name  18. 

Van  Schooten  133,   134,   138. 

Vissiere  18,  19,  29. 

Vlacq   268. 

Wada  Nei   114,    219,   220,  230, 

248,  271. 

Wake  Yukimasa  276. 
Walius  256. 
Wallis   154. 
Wang  Pao-ling  8. 
Wang  Pao-san  8. 
Wasan   i . 

Watanabe  Ishin  267,  268. 
Watanabe  Manzo  Kazu  76. 
Wei  Chih   112,   147. 


Westphal  33. 

Williams  5. 

Wittstein  197. 

Wu-t'sao  Suan-shu  9,   n. 

Wylie  10,    n,   12,   19,  49,  274. 

Xavier  255. 

Yamada  Jisuke  231. 
Yamada  Jusei  64. 
Yamaji  Kaiko  266,   268. 
Yamaji  Shuju   177,  181,  183. 
Yamamoto  Hifumi  176. 
Yamamoto  Hokuzan  274. 
Yamamoto  Kakuan   166,  176. 
Yamamoto  Kazen  243,  245. 
Yang  Houei  (Hoei,   Hwuy,  Hui) 

21,  22,  51,   116. 
Yanagawa  Shunzo   275. 
Yanagi  Yuyetsu  252. 
Yegawa  Keishi  259,  270. 
Yenami  Washo  64. 
Yendan  process    103,   129,  130. 
Yenri   92,    143,    150,    196,   200, 

212,  218,  225,  230,  238,  240, 

248. 

Yih-king  20. 
Yokoyama   136,   138. 
Yoshida  17,  44,  59,  66,  84. 
Yoshida's  problems  66. 
Yoshikadsu  180. 
Yoshio  Shunzo   266,   277. 
Yoshitane  64. 
Yosho   181. 
Yoshio  277. 
Yuasa  Tokushi  128. 


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